Is 'left' necessary?

It is not strictly necessary but it is perfectly appropriate.

The "left" implies that it was once someone's wallet, or that someone left it there for him to find. Wallets always have owners. The situation would be different if you replaced "wallet" with an item that typically doesn't have an owner; then the "left" would be required for context. At least that's what I think.

John found a wallet on the chair.
John found a wallet left on the chair.


John found a rock on the chair.
John found a rock left on the chair.

Unless the laws of physics have changed a bit since I last looked, rocks cannot place themselves upon chairs.