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?
Your transaction will delay maybe over 2 days dependent to transaction volume(i think it's risky).
but, "A transaction may be safely sent without fees if these conditions are met:
It is smaller than 1,000 bytes.
All outputs are 0.01 BTC or larger.
Its priority is large enough (see the Technical Info section below)".
you can check about transaction fees here:
https://en.bitcoin.it/wiki/Transaction_fees~iki