If it's the realistic kind where you just don't age, the statistical certainty that you'll eventually die in an accident, or to war or murder. Your odds of getting to the heat death of the universe without making backups is pretty slim.
If it's the kind where you're indestructible, you're highly likely to encounter someone who tries to bury you alive in a subduction zone eventually, because humans are like that, and then you get to spend eternity slowly moving into the scorching mantle.
The death of the sun will then eventually set you free into the gravity well of the sun where you'll live burning hot untill heat death of the universe. What to do after that is anyone's guess
Well, depends. The Earth is actually right near the edge of where the sun will expand to, so there's a chance the scorched glob that used to be Earth will stay in orbit. Either way, it will still be hot for a while, and you're ultimately stuck in something solid - be it a dead planet or a white dwarf.
There is such a thing as merciful death; it would not be good to be cut off from it.