If you set your wallet to pay zero transaction fee what happens? Is there more than one possibility.
It makes sense that no transaction fee means that you get no priority, but will the transaction always be processed eventually? If yes, what is the maximum time frame? If no, is there a maximum time limit before it bounces back? Could it be stuck in limbo for months?
You've received a bit of good advice and a lot of bad information in this thread. Please be careful of whose feedback you trust.
What happens if you send a transaction without a fee depends entirely on what wallet you are using, and what your technical expertise is.
As one example...
If you are using the blockchain.info wallet, they will wait a few days for the transaction to confirm. If it has not confirmed yet after a few days have passed, they will stop broadcasting the transaction to the rest of the network, and will show you your balance as if you had never sent the transaction in the first place. If you then send a new transaction that re-uses the same inputs, the address will be labeled in their block explorer web pages as having been associated with a "double spend".
As another example...
If you are using Bitcoin Core, the wallet will continuously re-broadcast the transaction forever until it confirms. You might get lucky and a generous miner (or mining pool) may quickly confirm your transaction, or it could take a VERY long time. There is no guarantee that the transaction will EVER confirm, although realistically I would expect that it would eventually confirm. If you become impatient there are tools and/or methods that can be used to manually get the wallet to stop broadcasting the transaction. If you use one of those methods the wallet will again reflect your balance as if you had never sent the transaction in the first place (although even then there is a chance that the recipient or someone else on the network might choose to re-broadcast the transaction).