There will be only a limited number of moves he can make following your checks, so the 3-fold repetition of position will occur sooner rather than later. Once it does press the offer draw button and the game is drawn.
Etiquette for perpetual check


Keep in mind, 3-fold repetition is simply the same position occurring 3 times with the same player to move, said player having the same legal options. They do not require the moves to go back and forth on 4 consecutive turns.
I've had endgames in over the board play where 3-fold occurs many moves apart, like I recall one case where I was White with a King and 2 Bishops, Black had a King, Knight, and 3 pawns (a, b, and c). The same position with Black to move occurred after White's 53rd, White's 58th, and White's 62nd moves, and I claimed stating 62.Bc2 and I claim 3-fold repetition, and it's ruled a draw.
So the original post is hogwash. You can't say the position repeated 3 times, but that it can't be claimed. It doesn't matter how many moves apart they are. If a pawn has been moved, then it's not the same position. The ENTIRE BOARD must be exactly the same, not just the checking pattern. Also, 3-fold doesn't have to be check either!
What's the etiquette for a perpetual check situation? I'm playing a game where I'm 8 points down but can keep the opponents King in perpetual check. If I do otherwise on any move I'll likely be mated, so I have to keep it going. I've offered a draw twice but they just ignore it. I've also commented that I believe we're in this perpetual situation. I guess they're hanging on in case I make a mistake. The three move rule won't apply because he's using a pattern that will prevent it.
What's the consensus on this? I'd have thought a draw would be acceptable.