Is it 24 hours after finish, or a set time globally? All it says is "daily" on the dungeons and I can't seem to find the exact meaning of that anywhere.
Some of my dungeons I finished Thursday night and now it's mid Saturday and they still say exhausted.
You have to refresh to pick up the reset. If you keep the game running in a browser, they'll stay exhausted. You can use this to your advantage if you want to clear out a few more dungeons, than reset the game to clear them out again.