Look at the Bo-ones.

[Formendacil]

Well, I don't know for sure about Denethor's bones, but someone must have moved the Palantir- indeed looked in it, because, to quote:

Then Denethor leapt upon the table, and standing there wreathed in fire and smoke he took up the staff of his stewardship that lay at his feet and broke it on his knee. Casting the pieces into the blaze he bowed and laird himself on the table, clasping the palantir with both hands upon his breast. And it was said that ever after, if any many looked in that Stone, unless he had a great strength of will to turn it to other purpose, he saw only two aged hands withering in flame.

How would they know this if nobody handled the palantirp?

I would take Tolkien's words to mean that nobody ever saw him, in life again. I admit that that is rather redundant, I mean it's pretty obvious that Denethor dies. Perhaps, though, this is just a more final, mythic, way of expressing it.

On the other hand, perhaps it means that nobody ever saw Denethor's ghost/spirit in times to come. Denethor being the sort of person he is, it wouldn't have been a huge surprise to see him come back and haunt Faramir, setting aside Man's departure from the Circles of the World for a moment.

I don't know what they would have done with Denethor's remains though (although I am pretty sure that they did something). I doubt they would have had a funeral service or anything similar for him, not with the War going on, and then the Return of the King, and the commencement of his reign. You wouldn't want a reminder of the old anti-monarchist at this point, would you? Perhaps there was a private ceremony of some sorts. In any case, I imagine that the remains were interred in Rath Dinen. Denethor was still a steward- and one who had done his job well for quite a few years. He still deserved to lay in his prepared tomb.

At least, all that is in my opinion.