Is money finite?

This answer is from an item perspective and will not mention TMs.

Because items in various locations respawn after one real world day (Moon Stones and Stardust in Mt. Moon, Nuggets and Pearls in the Underground Tunnel), I can say with confidence that it is impossible to run out of money. These items can be picked up repeatedly and sold to shops.

Based on what was discussed in the comments, no, money is not finite.

The move used to grind money (Payday) is now given as a TM just before the cave between Gym 1 and 2.

I havent checked, but I would guess that since its a TM, you can teach it to others than Meowth, and then grind the money you need.

Only thing that could break my answer (and I'm not there yet) is that there is no random encounters in the game. So at the end of all trainers, if there is no trainer that you can refight (If I can remember right, you can fight the elite 4 again and again), this answer is void.