My HTC One M8 is doing the same thing. It will not refresh when I open it, set it to push or set it to every 5 minutes. It appears that the only way to refresh it is to pull down emails while in app.
It worked when I first got it. Several apps have updated since then but no major changes. I did set a different ringer for notifications and that is when it may have started, odd. Calling AT&T. it appears that the AT&T Smart WiF had a bug in it also . . . Got an update and it seems to have gone away . . .