TL;DR: There will be some, but it will depend a lot on the initial situation.
Your ageless immortal is merely ageless. While that might eliminate some diseases that are caused by old age, it may not eliminate diseases by outside factors, nor will it prevent accidents, illness, and predation from taking their life. And make no mistake, something will kill them eventually.
We living things might be survival of the fittest in an individual sense, but evolution is all about survival of the good enough. It's about random mutations showing up and seeing if they breed true and help the species.
As a side note, I am making no assumptions as to species -- this should hold for anything from aardvarks to zebras. We, of course, are filed under H for human.
The initial impact will depend on a few factors
- Gender -- A female can only have one set of children at a time while a male can impregnate multiple females at a time
- Initial population -- Them being 1 out of 100 individuals will have more of an impact than being 1 in 1,000,000.
- Heritability -- How easy is it to inherit all or part of this individual's immortality
- Environment -- How likely is this individual to die of non-aging related disease/illness, accident, or predation?
The inheritance mechanics will be the largest factor in this set. As stated, if everyone descended from this one individual gains the full immortality, then it will eventually become the dominant trait I would think. What effect that has will certainly be based on what the species in. Immortal bunnies will have a different impact than immortal humans or wolves. Remember, they can still starve to death should their numbers exceed their environment's capacity to support them and they cannot relocate to an area that can support them.
Inversely, if they don't inherit anything outwardly, then they aren't getting anything interesting unless another random mutation or atavism down the progenitor's line makes another individual immortal. In which case, the whole process starts again.
Another possibility is that inheriting part of the immortal's genes gives them part of the benefit, but not the whole package. Some might be immune to age-related diseases, while other gain an increased theoretical lifespan though little defense against age-related diseases. Breeding between the lineages might wind up with another fully immortal member of the species to again propagate the gifts to.
As time goes on, your immortal will still have the urge to reproduce (as per the question) and will likely do so, either continuously siring children or having them. This means their presence in the gene pool will always be topped up, so to speak. This could definitely impact the diversity of the species. The impact will depend on how much the immortal's species manages to avoid inbreeding, especially close inbreeding.
But another thing your immortal has is experience. If they can teach their descendants, immediate or otherwise, a portion of their experience and knowledge, that will potentially give those youngsters an advantage in their life.
- For a prey species, that could be ways to avoid predators better.
- A predator species could hone its hunting tactics over the decades/centuries and pass those skills down.
- For humans or other similarly intelligent species, that could be more in life hacks, connections or money.
- Your immortal will also know the best ways to get a mate so should have no trouble reproducing should they want to -- another thing they can pass on if they desire their bloodline to continue.
What this could lead to is others trying to get the immortal to accept their children, even if they aren't the immortal's themselves. Depending on the intelligence of the species, this could lead to the immortal procreating less by being a surrogate parent for other's children, not like how aunts/uncles or grandparents can fulfill the role for us.
These immortal's cells do not age out so in theory, they could be cultivated forever. With an appropriate sample kept, cell samples could be continuously gown and experimented on over the years, allowing for a constant medical reference. As the cell shouldn't be susceptible to cancers from bad divisions, the only issues with the cells should be introduced ones provided the initial sample was healthy. This could prove valuable for medical research.
While this does not affect evolutionary pressures in so many words, medical treatments will allow for some to reproduce that might not have been able to due to illness or disease.
Likewise, if the immortal's status is known, they might become a sought after commodity for unscrupulous humans to serve as genetic fodder in hopes for immortal children. Their drive to reproduce might be hampered by persistent people that will not take no for an answer.
As has been brought up, they might also be driven from a smaller settlement once their status is known as they will be "the other" at that point and the (possibly jealous) mortals may force them out, meaning the immortal will have to find a new place to live. Then again, the opposite might be true if the society reveres its elders.