Is Pokemon Yellow money finite?

Almost; the Elite Four and payday are the only infinite source of money in the game (if you've reached them, of course). Later games all introduce some non-E4 method of getting endless money but in Gen 1 it's E4/Payday grinding or nothing.

Also, Payday is crazy slow for earning money, recommend just spamming the E4. There's really not much to DO with infinite money in gen 1 anyway.

No. Money is not finite. For example, Meowth learns the move pay day, which returns money back to the player after it is used in a fight.

Pay Day does damage and scatters coins on the ground with a value equal to twice the user's level for each time it is used. These coins are picked up afterwards if the player wins the battle.

Source

Also, you may fight the elite four over and over to get additional money.

No. There are ways to get money, even after defeating all trainers.

• You can battle Elite Four. This is much more effective than other methods on this list.
• You can use Pay Day in a battle, and it will give you money. It's level-up move for Meowth, but it also can be learned from TM16 which Pikachu can learn.
• You can earn coins in the Game Corner by playing, and sell the prizes. It's hardly optimal because it takes 3300 coins to get 2500 money, but it's possible.
• You can move items from another game (even one that you just began) using Pokémon Stadium 2, and sell them in-game.

Also, it's possible to abuse glitches.