Obi-Wan Keeps it.
I think Obi-Wan will keep the lightsabre as a sign of respect for his master Qui-gon. When Jedi Apprentices are constructing their own lightsabre they are supposed to give it a similar look as their master's lightsabre as it shows respect, so Obi-Wan keeping the sabre would be just as good.
Re: Obi-Wan Keeps it.
Obi-Wan will definately keep the saber whether he uses it in the future or not. I don't think it was burned with Qui-Gon's body because GL would have made a brief point of noting it....however I recently found it (which I confess I did not know before) that in ROTJ, Luke burned only Vader's armor, NOT the body. What happened to the body ? I would think it would have disappeared like Ben and Yoda if Vader had made his peace with the good side before he died on the flight deck. Any ideas out there or am I way out on the rim of the Anoat system ?
Re: green lightsaber
I just read part of the novel and in the scene where Luke burns Vader it says that "Vaders lifeless body lay there still" or something like that. Therefore, Vader did not disappear and he was burnt just as Qui-Gon was. Also in the novel it says that after Luke strikes Vader down on the railing he drops his lightsabre. So Vader's lightsabre was not burnt with him.
Even if the novels do not count as cannon I think Lucas would have approved this to be in the novel and therefore Vader definately didn't fade away.