Let's say that you have a person who's, if certainly not immortal in the general sense, biologically immortal, as enumerated in this question - not only do their telomeres never shorten, they're also-near-immune to cancer. This means that, while they're going to die eventually, due to a variety of factors, they'll also live for a very long time if they don't have some kind of accident. They can be physically killed, but their body will never break down.
Anon's answer to that question pointed out that:
The other problem here is that the human body does not retain the ability to replace structures. If your adult teeth are removed or badly damaged, there is no biological mechanism in place to repair or replace them, for example. This will lead to another answer to your question, along the lines of "what life-critical part will wear out first"? Specifically in females who birth children, depleted-bone-calcium is a contender - if that's still a part of your future, population control will be a consideration.
Assume that this hypothetical long-lived individual lives for a thousand years. I'm not sure if that's actually how long they'd live, but let's use it as a benchmark.
So, aside from adult teeth, and bone calcium, what other parts of the body will irreversibly wear out in the span of a thousand years?