Fawkes was Dumbledore's pet - and VERY loyal to him. It is uncertain whether or not a phoenix can only have one owner or if he can have more than one. If it is true that he can only have one, then Fawkes is probably gone for good as Dumbledore has died and is not coming back in any 'live' sense of the word.
Otherwise, you have to wonder how someone ends up with a Phoenix as a pet. Does the pet chose the wizard type of thing. If Fawkes chose Dumbledore, might he not come back and choose Harry?
Could Fawkes have gone to Godric's Hallow? How old is the phoenix? How many burning days has he had? I don't believe we have seen the last of Fawkes.
So many questions... not enough answers...
Have you reread Harry Potter books 1-6 recently? See how well you remember the facts with this Harry Potter Mega Quiz.