For an elf-like race whose people do not age beyond physical maturity (the accumulated DNA damage associated with aging does not occur), what would prevent overpopulation?
This elf-like race is fairly similar to the stock fantasy elves. Similar to humans, but more elegant, healthier, etc. It should be noted that these people are fully capable of dying, but aging is never the culprit.
Aging is not necessary to maintain their low population. No population can grow without limit, and in the animal kingdom, most animals die before they would otherwise die of extreme old age.
These are the things that limit population growth.
Food stocks. Regardless of your will to reproduce, if you don't have enough food, you can't grow new elves. Advances in crops and hunting can limit this.
Predation from other species and their own species. Other species need to eat, and elf flesh is delicious.
Environmental toxins. The elves, or their presence, may damage the environment in various ways with their waste that leads to poisoning or death.
Disease. The greater the population density, the faster a disease can spread. If you get enough people more diseases will spread.
Habitat availability. There is a fixed amount of wood, caves and such for elves to live in. If too many are born, they will have nowhere to shelter from the weather and predators.
Climate extremes. The limited resources of the environment may limit even more sharply at certain times of years, when rivers flood and storms come. This may limit the population that can be maintained.
This will be true regardless of how fast they breed or how long they live. They'll hit caps in their growth.
They will be stronger than other species.
Species have reduced fitness due to aging. See the table below. Focus on the orange column, which shows the percentage of deaths where old age helped cause it by species. The other columns show the confidence of the data and other statistical features, but knowing their meaning is not especially important. 
Animals get physically weaker with age. The contribution of aging to death ranges from 3-70% based on this table. The elves will have a major competitive advantage over other species. This advantage will be greater in safer environments, and less in more dangerous ones.
They would live long, but procreating a child is a very rare event for all couples.
That should balance the lack of aging, if deaths due to other reasons are not too frequent.
Older elves have a lower libido, but more importantly, producing offspring drains the soul or "life force" of the parent. See the accepted answer of https://scifi.stackexchange.com/questions/91588/why-dont-elves-have-more-babies for a thorough explanation. Essentially having too many children can kill you.
You could modify Tolkien's system to have the "life force" recharge, albeit slowly, over time. That way elvish reproduction wouldn't stop entirely if a younger generation is wiped out in some war.
A way birth rates could be reduced would be to have a longer periods for the child-bearing process. Meaning that the development of the child in the womb could longer than a human's 9 months, and the physical recovery time for the child-bearing elf could be many years. This would not necessarily mean that an elf who gave birth would be incapacitated during that time, but that they are effectively sterile during that time period, and thus reducing the rate at which elves can reproduce.
As in many worlds where elves are used, they often have close ties to magic, nature, and the such. It could be possible that there have always been approximately the same number of elves at any given time on the world, and that could be because when an elf dies their energy returns to the world, and is used by nature/magic/whatever to create a new elfling. These elves could still sexual organs if you want to include that with your world, but they do not have to be used for the purposes of creating offspring.
In the Drowtales web comic, almost all characters are elves and thus are able to live forever. Many of the major characters have lived for over a thousand years. But they don't ever get a case of overpopulation because the underdark is probably the only fictional place that kills more named characters than Westeros.
Young male elves have access to the internet. They are also addicted to DotA, League of Legends, Battlefield, Overwatch, Call of Duty and Fortnite (or whatever game kids are playing by the time you are reading this). You'll be hardpressed finding better contraceptives.
Ten years of foreplay
Maybe, very specific conditions need to happen for ovulation (and likely conditions for sperm production).
If, gestation takes a long time and the children are helpless for decades instead of years, natural selection would favor those who produce children only when conditions are right to produce a child that will survive to bear children. This only really works if they developed in an environment where there were no large threats to the population (to prevent the produce many to ensure that some survive). That doesn't mean that there aren't threats but they should be threats that are manageable once the child reaches a certain age.
This avoids spending energy and resources on a child that might not survive (it's not like the parents will run out of time).
This also makes half-elves very unlikely. There are few humans who would go to that much work. So a half-elf would either be the product of a great deal of dedication on the part of the human (and a sense of urgency for the elf) or extreme luck to find an elf at just the right time for the conditions to have been met but who is currently without a partner.
This type of elven community would treasure its children since each child is the product of devotion.
Unless there is a separate community that elves with children move to (and it might make survival sense to have such a community in the most protected area), it would mean that elven childhood friends are likely the result of planning and coordination of the respective parents.
None of this means that the elves don't have sex, it's just child production that takes 10 years of work. Casual sex may just take a month.
On the other hand, they make like bunnies with no consequence. It depends on which way you want to take it.
My guess would be a either a Cultural solution or constant warfare. Perhaps they have decided upon a system of procreation that limits how many people procreate and or when.
Cultural
Only the rich are allowed to have children, or there is a tax on having children.
Maybe it is viewed as a faux pas to have children under certain circumstances.
After a person reaches a certain age they are sacrificed for the greater good of the clan.
Once a year there is a raffle...
As for warfare, that is fairly self-explanatory, the race is constantly killing each other.
A few more possibilities would be that they become prone to sickness and disease at some point, and so although they do not die of old age, many of them do end up dying of "Natural Causes" eventually.
Due to their body shape or constraints, Elven mothers are only able to give birth once in their entire life time. Either they are permanently unable to give birth again (recommended scenario, so that kids get a mother), or they die from giving birth (fathers full responsibility to raise the kids): so females try to avoid getting pregnant as much as possible. Population growth is stable thanks to twins or triplets.
To biologically support the inability to give birth again, you could have menopause triggering as soon as you give birth.
I agree with @L.Dutch and @Nepene Nep answers. They would have low natality rates and the environment may be dangerous or the resources scare.
But if you like a bit more "mystical" explanation, here you have.
Do you know about ants and their colonies? Ants have "roles", for being exactly 4, fighters, explorers, gatherers and another role that I forget. Some look for new sources of food, other gather and bring back food to their base, and others protect those ants. But, what if a creature starts killing all the gatherers? The colony will start staving very soon.
Well, they have statistics... really. Ants use to move in specific paths or road, very close each one from another. When an ant touches another, they exchange chemical substances (I think hormones). They have 4 chemicals, one for each role. Each time they "smell" a chemical, they remember it and take note.
This is when things become interesting, each ant is always tracking a summary of each role and its amount over the day. So, when an ant note there are a lot of e.g: fighters but almost none is e.g: gatherer, they automatically change of chemical and role: they become gatherers.
Elves do something similar. They are always instinctively tracking the population of their communities. When this population becomes low, they instinctively start having offspring, when the population is very large or there are only a few resources available, they instinctively stop having offspring.
Limit the number of eggs per female elf
A female human is born with all her egg cells in place, I think there are around 400 of them. She ovulates once per month, losing an egg, regardless of her behavior, and when they are gone, she enters menopause and cannot bear children any more.
Have female elves be born with only a handful of egg cells. To ensure a celibate elf doesn't waste all of them, make her only ovulate after mating - this doesn't stretch the imagination too far, since it is present in many Earth species, including mammals. As a side effect, she'll also thank you for not going through a monthly cycle. If you can introduce some kind of voluntary ovulation too, or easily controlled ovulation (for induction, you need both a mating event and consumption of belladonna leaves), you also create a good starting point for a society with high gender equality and a strong tradition of female scholars or artists, which is usually a plus in contemporary fiction.
Don't limit your elves to only 2 or 3 eggs, since not every induced ovulation results in a fertilization, and not every fertilization results in a conception. There are also unsuccessful pregnancies, elves whose children died young, and natural events which decimate the race, like wars and diseases. So, for a population that survives in the long term around some stable attractor, give them somewhere around 6 to 10 egg cells per female, or a bit more (15-ish) if you are OK with a more volatile population that is better at rebouncing from adverse events, but also pushes against natural limits in good times, creating something like a very slow Lotka Volterra cycle.
Fix the number of elven names.
I have a better solution. This requires elves to be consciously selective when they get pregnant. Have a social construct that the names given to them is limited and fixed. Thus, unless a member of the society dies, no one makes a baby. When someone dies, elders decide who to make a new baby to fulfill that name. Will work wonderfully with an elven society.
If an "accident" occurs, both parents are exiled, their names are taken, and the newborn will get the name of one of his/her parents.
Simple: absence of involuntary pregnancy.
Arguments are fairly convincing that this would prevent overpopulation in humans; generally people don't voluntarily have excessive numbers of children unless there are social and economic systems in place that outweigh the costs/burdens of having additional children with advantages, so in the absence of of such systems, the main cause for people still having lots of children is lack of reproductive autonomy. Disincentives would be even heavier in a species where raising a child takes many decades of work.
So, either give your elves natural control over their fertility independent of sexuality, or give them universal awareness of and access to contraceptive technology. Awareness and access are very plausible in a society with low population, high education, long lifetimes, low poverty, etc. that are usually ascribed to elves.
Lunker elves.
A lunker fish is a very big very old fish. In a lot of circumstances these are the fish that do the most reproducing. Their genes are tested by time and their large size means the males outcompete smaller fish for mates, and the females can dump lots of resources into eggs.
So too your elves. They are not reproductively mature until age 1000. Then they start making babies and lots of them. Even though they don't age, they do die from wars and accidents and sometimes disease and most do - 1000 year old elves are not common. To make it to 1000 you have to have genes for long life and also disease resistance and good sense.
This means that many elves in a community will be siblings (or half-siblings; the elves are open minded about these things). Without children of their own, these half siblings are available to care for their younger half siblings and so the old elves have one baby after another, and the babies are raised communally.
As regards sexually active that depends on your story. If you want the hot sexy Legolas types you could have sex be the glue of elf society even among those who are reproductively immature, like with bonobos. Or your elves could be elfin lithe presexual 10 year olds for a millennium.
I also like that the old reproductive elves are not obligately one sex or the other - over a period of a year or so they can shift from male to female and later go back. After bearing several children, a female can shift to male and pay visits to some of the old elves in other communities. A pair might have many children together, each one of the pair a father to some and a mother to others.
Biological imperative to return to place of birth to reproduce
Perhaps similar to salmon that are driven to expend tremendous energy without great chance of success to return to an upstream birthplace for spawning, a particular elven race must return to a specific location in order to procreate.
The place of birth was left behind as the elves evolved into higher planes of existence and it is now seen by those old enough to procreate as an inhabitable cesspool that can barely be tolerated. They have no choice but to return there if they wish to have children however, because the needed environmental condition of your choice is found only there.
Most of them simply choose to go childless in order to avoid the distasteful experience of the location.
Elves reproduce slowly. Humans think elf ears are good luck. Elf scrotum is considered to be a powerful aphrodisiac. Elf gall bladder is believed to cure syphillus. Thus the population is not just controlled, they are driven to near extinction.(dim witted, short lived,rapidly reproducing humans however thrive)
Fetus absorption or other means for coupling birth rates to population size and/or resource availability
This might be a bit too fleshly-icky for beautiful, pure, highly moral elf races, but I am still posting it as an alternative since it is a very realistic explanation and you might need that. I am posting it as a separate answer, because it has some fundamental differences from my other one.
Fetus absorption is something that already exists in several species on Earth, I believe mostly rodents, but maybe also some marsupials. Females of childbearing age get pregnant frequently. But when they are under stress, especially food related stress, they don't give birth, and don't abort the fetuses. Instead, the fetus tissue gets absorbed by the mother's body. Now, you probably don't want your elf mothers to go around as starved as wild mice in a desert to prevent them from overpopulating your biome, but you can easily think of a limited nutrient which is needed for carrying a pregnancy to term, but its lack doesn't reduce the wellbeing of elves in any other way.
You can also forego nutrients and use pheromones. There are species where the presence of high ranking females inhibits ovulation in low ranking females, although it doesn't stop it completely. To have a more egalitarian society (if you want that!) you could create a world in which a high elf pheromone density in the environment inhibits pregnancy or birth.
This strategy has the advantage of being adaptable and self-regulating, tending to return to the same population density over time, no matter what shocks you introduce to the system. When you have reached optimal elf density, births get extremely rare, just enough to sustain the population. If some event suddenly increases the density (e.g. half the elves' habitat is destroyed, say by an earthquake, and elves overcrowd the other half), the birth rate drops. But if you have a sudden expansion, say elves colonize a new planet, you get a very quick rise to stable numbers again, unlike strategies which rely on very slow reproduction, or limit the number of births per couple or mother. It is also evolutionary sound, as opposed to purely cultural ways of population control, and can be made quite pleasant to the elves themselves. It can even give you interesting hooks for story arcs (a young elf running away to an inhospitable land with her lover because she knows that at home, her chance of conception is towards zero, for example). And it will work both in prehistoric elves living in the savannah (forest?) where food access is scarce, and in developed elves living in a world with an overabundance of food.
1) Low natality rates.
Having a kid is hard for them, and since they live extremely long lives (like forever unless killed) they don't need to hurry.
2) They only have sex with their soulmate
Removing meaningless sex from the equation is sure to reduce the procreation numbers.
3) They can control when they want to have a baby
How? magic? maybe just standard but better-since-they-are-elfish condoms? kinda depends on your world).
Easy+free contraception is good to stop their numbers from growing too fast.
4) They understand having an elf-baby as the ultimate responsability
Therefore they don't have a lot of them.
4b) Maybe... Elf-babies require a lot of time to get to maturity, so it's a big commitment from their parents
Low resources. Humanity's real-world ability to support billions of people has only been possible since industrialization. In stories, elves are usually pictured as living in the forest off the fruits of the earth, and hunter-gatherer techniques don't provide a lot of sustainable food. Even if they farm on a large scale, farming is still a very inefficient process without heavy machinery (probably more so for elves, who are usually depicted as being slender and agile rather than big and muscular). In the end it doesn't matter how long a lifespan they have if they can't produce enough food or other resources to support more than a few thousand individuals.
Whatever magical trade elves made in their history for ageless beauty might well be a curse that results in their eventual extinction. Agelessness just isn't very desirable or adaptive as far as reproduction or evolution is concerned. If one race can turn out 10 children in 15 years vs another one that manages one child every 150 years, you should be able to guess which one is evolving and adapting one hundred times faster.
Insert fast breeding humans that compete for the same resources into a population of slow breeding elves and expect no elves after a dozen generations. Insert fast breeding elves into a population of slow breeding elves and witness population boom-bust famine cycles. Insert fast breeding elf eating predators into a population of slow breeding elves and get no elves in just one generation.
Infinite physical space
We live on a finite globe, it does not have to be so in fantasy. I don't know if you can envision an infinite planet easily (I'm having problems with it right now at least). If you can make the tradeoff to not have a universe as we know it, then you can skip celestial bodies and just have a plane, those can be imagined as infinite more easily.
It also places some 'perspective' boundaries on the framing of the story. With infinite resources there is now less opportunity for conflict. This can of course be solved, but that's for another question I think. (Geographic elements could be one answer)
The story will also probably be nudged to have a more 'local' flavour rather than world-ending-and-global-storyline. If you have infinity as a major theme; then a grand story of great importance to the world could be maintained.
For a setting I developed, I had biologically immortal humans (not elves, but that shouldn't stop you).
The main mechanism I used for population control was that the value of human life in that society became very low. This translated (among other things) to very harsh punishments for any crime: even simple theft invoked capital punishment.
Another factor was that the island was invisible from afar, and its location was constantly changing (so you can probably suspect that it was a bit more than just an island, but that doesn't matter here). This mechanism made it improbable for anyone searching for that island to have any success in finding it, including people who had been on it and left at some point. So that society had outward mobility only (which doesn't do a lot, but it contributes in a small way to limiting the local population).
Could depend on the society, if they are militaristic where officers and non-combatants are not likely to die but there is enough going off to war to die in combat or have something like in the more fictionalized Sparta with kids having to fight to the death to prove they are worthy of living. You could also have a society advanced enough to know that population would be a problem and to stop if they have form of birth control. You could also have something like a high rate of birth defects that kills the child at birth or at a young age.
I looked into cause of death in humans, then took out all of the old age and disease causes to see what that would do to the lifespan.
All other things being equal it extends to an average of 1700 years. (Since then I've seen figures up to 3500 years. Hey, this isn't Actuaries.SE) But there are two additional factors:
Since accidents and murders are random, the distribution has an exponential distribution. In effect a half life of 1700 years.
A second effect, that of maturity comes into play. With humans the dangerous decade for accidents is about 14 to 25. Males in particular do not evaluate risk well, and as a consequence there is significant mortality in this age group. (Suicide and car accidents...)
I know even sensible people in their 40's will still engage in hang gliding, rock climbing, diving. While they have survived the stupid decade, they are still taking a riskier decision.
The net result will, I suspect be that older people will be more cautious. The longer you live, the more cautious you are. This will have the effect of lengthening the tail of the curve. To use the radioactivity half life model it will act like a mix of shorter and longer lived isotopes.
A: High mortality in the young. For every 100 kids, only a handful make it maturity.
B: Culturally they decide that someone under X years old isn't mature enough to have kids. Set X to be a significant fraction of their average life span. (Historically, humans pair and form families about 1/3 of the way through their lives. When few people reached 50, most were married by 16-18. Now we live to 70-85, and we have first kids in our late 20's. It's not perfect correlation, but it's there.)
C: The secret to their immortality is cultural, not genetic. They take a treatment for long life, and the consequence of this treatment is sterility. This would encourage early childbearing, especially if the treatment just prevented further aging, but didn't make you young.
Make successful pregnancy difficult. Biologically this can come by low fertility rates, high sterility rates, and even when they van get pregnant, they suffer from a high number of miscarriages or stillbirths (early and late term fetal death). Along with their being a sentient species who understand the logistics of overpopulation vs resources, and low adult death rates, they settle into an equilibrium of population rates that does not grow beyond what is needed. Being biologically immortal they would not get the same human biological clock effect or the desire to replace dead relatives.
Side effect, twin and triplet preganacies are common.
Elves need space.
Overpopulation does not necessarily result from lack of resources. Simple population density can be lethal. The classic studies were performed by John Calhoun, in which he provided an enclosure for mice and rats with unlimited food and water. In theory, the population should have stabilized. In fact, in every case they crashed, with the entire population dying off. See here for a quick overview.
People don't seem to have the same built-in constraints, but elves are not human. Posit that, in this aspect, they behave like mice or rats. Let's say that the pheromones given off by other elves trigger bad behavior if the levels get too high, or whatever mechanism you like. Or don't even bother with a mechanism, It Just Is.
So, the elves have learned (the hard way) that they must not let their population density get too high. They do whatever is necessary to limit their population.
