This is actually a scientific question that has already garnered a lot of philosophical thought. Many have noted that immortality might be possible in the next few hundred years or so. So some investigation into its implications have already begun.
I suggest you do some light reading on the ethics of immortality:
Pros and Cons
JStor scientific article
However here are some key points that you may want to be aware of.
A. Never aging does not mean an end to death. I believe the estimated lifespan of someone who never dies from old age (but can still die from car accidents, war, muggings, viruses, etc) places humans at around 1000 year life expectancies. So while over-population will become an increasing problem, it is not like there will be infinite humans that keep piling up and never go away.
B. Lowering birth rates can counter overpopulation. Are you aware that all first world countries have declining birth rates? Why is probably a cobimnation of technology, passtimes, cost, and so on. However it is not a stretch to believe that if all humans are uplifted to the lifestyle of first world countries (and the lifestyle of the future), that birth rates may drop off drastically. Perhaps to the point that birthrates may only be a little higher than the estimated death rates of humans.
C. Space Travel. With the advent of space travel, and offworld colonies a possibility in the near future, it is then plausible to think that excess human capital can move offworld. If this is the case, then we may want the historically high human-birthrate. This is because there are bound to be a lot of accidents in space and on hostile planets that would lead to loss of life, and the need for colonists. So this may be another way to deal with an overpopulation issue.
We just need to maintain context, and understand that immortality, being a technological advancement, may be accompanied with other technological advancements like space travel.
D. Social Safety Net Programs. I don't think that it is a stretch to predict that almost all of these would disappear (at least as they are related to work). If someone doesn't age, and can work their whole life, then why do they need to be looked after into old age? They don't. Social safety net programs will in many ways becomes obsolete. The ones that may stay, are a universal income, or food stamps to prevent people from dying.
E. Natural selection. If you understand natural selection you will understand that natural selection never stops. If all humans never died (a more extreme case than will be the reality) then evolution would never stop. The only thing you may get is increased interference for a while, as newer generations breed with older generations and genetic change has a few rollbacks in the march forwards.
Overt countermeasures will most likely not be required. Social countermeasures will come into place on their own, and future opportunities may mediate all foreseen problems.