The minimum recommended fee is BTC 0.0005. Below that, all the other nodes might reject your transaction, which is what is happening in your case.
I tried to send with no fee but the send button was grayed out so I couldn't click it...
So it's forcing me to set a fee, but I was able to set it to 0.00000001 and send it successfully.
I've noticed that too. It doesn't require non-zero value, simply a value. Zero works fine.