My problem is similar to this recent post about the S5:
MMS Problems with Samsung S5
It started this morning (Wednesday, 6/11/14). I have sent and received MMS messages seamlessly until about 8:00 this morning. Now, if I am connected to a wifi network, but do also have data enabled, I cannot send or receive MMS messages. When sending, it just hangs and doesn't send until I disable wifi. When receiving, I get a message wanting me to click "download", but when I do, nothing happens. Turning off wifi makes both messages go through immediately.
I understand that MMS messages do not go through wifi, but, as I said, until this morning they could go over the data network as needed even when I was connected to a wifi network. The change was sudden, and I'm not aware of anything that changed with my wifi connection or my phone, though my apps do typically auto-update, and I don't know which, if any, may have auto-updated this morning.