Transaction fees are required by the network if your transaction has any one or more of the following attributes:
- Transaction size is greater than or equal to 10,000 bytes
- Any output is less than 0.01 BTC
- Transaction priority is less than or equal to 57,600,000
If you create a transaction without a fee that has any of those attributes, nearly all peers will refuse to relay the transaction. So it won't make it off your computer to the network.
Assuming your transaction passes that "spam test", you can attempt to send it without a fee. It may take a very long time for any miners to include it in a block (generate a confirmation). If you want to try, I believe that the blockchain.info "custom" transaction creation will allow you to specify no fee. I think the Bitcoin-Qt wallet will also allow you to set the transaction fee to 0. I'm not sure about Electrum.