Are commas needed?

In my opinion, 'yes'. Otherwise, the sentence makes no sense and is awkward.

Strictly speaking no (I think), but I prefer them to be in.

Indeed they are required as they separate two distinct thoughts. In other words, "My friend is a close friend of my father" is the primary thought whereas "whose father is a doctor" is a secondary thought.

Without commas -- My friend whose father is a doctor -- the sentence would, strictly speaking, indicate that you wish to distinguish this friend from those friends of yours whose fathers are not doctors.

With commas -- My friend, whose father is a doctor, -- serves to describe your friend to some extent, rather than identify the friend. It's a subtle difference, and would probably be lost on most readers (that is, the sentence without the commas would probably just be read as a sentence that should have commas and doesn't), but it is a difference nonetheless.

Just to expand on patiodog's answer a bit...

Commas are not used when the clause is essential, that is, when the clause uniquely
identifies the word it modifies. On the other hand, commas are used when the
clause is nonessential; the clause describes but does not uniquely identify the
word it modifies.

Hey, good words. Nobody never learned me no grammar.

Does it make any sense with or without the commas?

My friend is a close friend of my father.

Maybe I'm just a bit tired.