I had this problem too when I switched to a Note 3 back in Novemeber 2013. MMS and Group messages would not work as long as I was on WiFi. If I turned wifi off, they worked fine. I was able to find a solution that 100% fixed this problem for me. Im not guaranteeing it will work for you and your phone on your specific carrier, but this is how I fixed it on my phone.
It involves the APN (Access Point Names) settings in the Networks options. What I did was went in there and copied down all of my Carriers Default APN settings. I then created a new APN. I typed in all of my phones Default APN settings, but changed 2 lines. And it has to do with the "APN protocol".
For APN protocol I changed it from just IPv6 to IPV4/IPv6
My phone has "APN roaming protocol" too just below, I changed that line to the IPv4/IPv6 too just for good measure.
After I added the IPv4 to APN protocol, eureka, I was able to send and receive MMS and Group Messages while connected to all WiFI without it failing or saying "push to download" rubbish.
Maybe this will work for you all? It did for me! I found this solution after 2 months of searching for solution from an obscure xda forum post. Good luck!
Edit - I forgot to mention that if anyone is having this problem while using T-Mobile for their Carrier, you have to turn off "WiFi Calling" or select "Cellular Preferred" for this to work. Otherwise you phone will still be trying to send/receive MMS through WiFi and it will fail due to those messages only working through an active Data Connection.