I have a game currently which is King and Rook versus King and Bishop. My understanding is that this is a book draw but my opponent has asked to continue, which I respect and is perfectly fine. However, at what point does it become a draw, could we continue for ever?

after 50 moves by each player where a piece is not taken or a pawn is not moved, if i remember correctly.

50 moves by both players with no pieces being taken and no pawns advancing.

OR

3 moves which have the exact same positions, does not have to be consecutive.

Do I have to claim it, or does it happen automatically?

it has to be claimed.

Well, not necessarily so[!] - assuming that Black has the bishop, and as long as Black's king is not confined to a corner square of the same colour as the bishop, nor subject to a back-rank mating threat - this is not what you'd apparently call a 'pure book'-type draw, but in layman's terms, a 'theoretical'-type draw. 

More often than not however, this type of scenario (either with bishop or knight vs rook) often leads to some type of draw, either through 'repetition of position' [or 'threefold repetition' rule/law], or via the '50-move' rule/law, but more often through 'agreement' - and rarely [though remotely possible] through 'stalemate'.

Theoretically, without any outside [USCF/FIDE/in-house, etc.] intervention of specific [draw claim] rulings, in most [specific] instances of bishop/knight vs rook endings, a chess game of this example could in fact last forever, forgiving player deaths, time itself, and I suppose the laws of the universe as well.

And that's the way I see it! ...