I posted a little about this on a similar thread at the Indy4 table, but I'll expand on it here.
We've already seen from the movies that Indy can survive any danger, not because he's invincible, but because he's determined. He gets injured, but he keeps going. He seems to be defeated, but he keeps going. Even if he got seriously wounded rescuing someone or doing some other heroic deed, he would remain determined to survive. The only way he could die in action is to loose that determination, and that would just be a sad way for him to go.
Another option is that he would be executed by someone. This would be different from dying in action, but I think the determination factor would still apply. Unless he gave up hope (which I don't imagine him doing) he would remain determined to escape and live, and that determination would make it possible.
Then there is the chance of sickness or disease. There are many determined people who have succumbed to death by disease so that is a plausible cause of death. But its not very heroic. Indy should not waste away in a hospital or nursing home somewhere. This is probably the weakest of my arguments, but that's just the way I see it.
It is my opinion that Indy died of old age/natural causes. He lived to a ripe old age and continued to be active as an old man. He went to bed one night, and just didn't wake up the next morning. We've seen from The Young Indiana Jones Chronicles that George Lucas thinks he lived to at least 92. In fact, it's my theory that Indy died sometime in 1993, since that's when they stopped using the "Old Indy" bookends on the Young Indy episodes.
It may be a morbid topic, but how he died should be a reflection of how he lived his life and what made him who he was, and those things would not change, even at his death.