I'm in the shrine with a Flame Atronach but my weapons have no effect and it won't let me exit the room.

You mean the Mount Kand shrine? You have to answer his riddle, and the riddles of the next two atronachs(Either check a walkthrough, or read the Yellow Book of Riddles in-game or at the Imperial Library website). If you kill him - or the other two - rest a day or two and he'll respawn.

It just started attacking me right away. It's at Shrine Assemanu.

Ahh, Sixth House shrine. Got it. Are you there for the Robe of St. Roris, or just there to be there? If you have any enchanted weapons, they should work. If not, use a destruction spell. If you don't have one, use Almsivi or Divine Intervention, or recall to get out of there. If you don't have any of those, I suggest reloading from a previous save and waiting until you've got a good defense.

Thanks.

Flame Antronachs have a weakness to frost, so enchanting a weapon with frost damage or creating a spell that does frost damage would be a good idea.

Oh, right. I forgot that they're immune to fire. Makes sense, doesn't it?