We obviously see Fawkes consumed by fire and then reborn from the ashes in Chamber of Secrets. If a Phoenix lives a natural life (i.e. isn't killed by something) will it continue to burn and be reborn infinitely, or will the Phoenix eventually die from natural causes (i.e old age, etc)?
I'm not sure how canonical this is, but:
HP Wikia "Immortality" says:
Whenever phoenixes die, whether from old age or something like a Killing Curse, they always reborn from their remaining ashes, technically making them immortal. They are so far the only living beings who possess natural immortality. They are also the only creatures who defy the absolute law of that nothing can truly bring back the dead.
Phoenix Wikia also says "Hence, phoenixes are immortal".
I don't recall any canon material contradicting that, but in all fairness none that proves it (e.g. forever regenerations) either
Though this is slightly out of scope of the original question, it's 100% certain you can NOT kill a Phoenix with Avada Kedavra, since Fawkes took one in HP5 to save Dumbledire's life, and was reborn:
"Fawkes swooped down in front of Dumbledore, opened his beak wide, and swallowed the jet of green light whole. He burst into flame and fell to the floor, small wrinkled, and flightless" (src: HP5, page 815 US book).
Out of Universe:
Lists a variety of myths, none of which seems to indicate ANY way of permanent death for phoenix.
Back in-universe, I can think of a couple of PLAUSIBLE ways to try and kill the phoenix, but with absolutely ZERO information in the canon AFAIK as to whether they are gonna work:
Have a Dementor kiss it
Have it be thrown through the Veil (Ala Sirius)
Place it in a vacuum (No oxigen => No burning, plus not enough energy for rebirth)
If the ashes actually play some active role in the rebirth (as opposed to merely being around an already re-born bird), scatter the ashes right after the burst.
Dibs on the questions based on those ideas! :)
It does not appear that Phoenixes are immortal in Potterverse:
The phoenix lives to an immense age as it can regenerate, bursting into flames when its body begins to fail and rising again from the ashes as a chick. Phoenix song is magical; it is reputed to increase the courage of the pure of heart and to strike fear into the hearts of the impure. Phoenix tears have powerful healing properties.
Fantastic Beasts and Where to Find Them - page 32 - Scholastic Books
I figure 6 weeks is long enough to wait for someone else to look this up, as per the comment I left for DVK.