One of the seminal philosophical works that inspired debate on this topic was Bernard William's The Makropulos Case: Reflections on the Tedium of Immortality. Williams argued against the desirability of immortality, and his basic argument was as follows. He reasoned that what makes life meaningful and worthwhile are the deeply rooted "categorical" desires, projects, goals and commitments, etc, that we tackle and pursue. These can come in many flavors and forms, but Williams argued that it is a non-contingent fact for beings like us that these categorical pursuits are finite and hence subject to being exhausted, similar to how one could exhaust a rolodex. So he reasoned there would, out of necessity, come a time during an endless existence where life becomes intolerably tedious and not worth living.
All of his major premises are debatable. But even if they are granted, one objection that can be raised is something already touched on by FMF and twhitehead: if the immortal being does not have total recall on his categorical pursuits, then even if they are exhaustible, they could reoccur without tedium. It would be like coming back to the start of the rolodex and not having recall that you have already been there before.
With respect to contingent versus necessary immortality it is worth pointing out that much of William's essay was focused on a contingent variety, even though his argument was not really explicit toward one or the other. That is because he focused on the fictional case of Makropulos, who had the option of taking the elixir every few hundred years or so. I think necessary or essential immortality is the really difficult case, but it is still worthwhile to consider the contingent variety. The contingent case is like the necessary case but with the added option of ending if things go badly. So if you find reason to declare the contingent variety undesirable, I would think that declaration should transfer also to the necessary variety. Of course, if you find the contingent sort desirable, it would still be a further question about the desirability of the necessary sort.
I think the really difficult thing about this discussion is that our judgments of (un)desirability are typically made within the scope of recognizably human terms; and immortality is not recognizably human. If we try to make it more recognizably human and more fully described, then it is difficult enough trying to make it a conceivable thing, let alone a desirable thing. This is one criticism that gets lobbed at some theists, that their notions of everlasting life in heaven is a bunch of underdescribed handwaving. At any rate for some of these theists, humans already possess immortality whether we like it or not. It's just that the conditions of eternal life can vary dramatically as represented in the two extremes of heaven and hell. Indeed, it seems there is nothing conceptually about immortality that makes it immune from poor or even insufferable life conditions. Being somewhat risk averse, I would think that pulling the trigger on taking the elixir without having very strong assurances that constrain the life conditions would be an enormous risk.