OK, an update to my saga.
My iphone 5 was working fine until the 6.1 update, and then started running hot, and burning battery like crazy.
I tried several suggestions I read online, which didn't work, and then took the phone in to the Apple store. The tech deleted my Exchange account and created a new one, which seemed to fix the problem - for almost a day.
Then the same problem started all over again. I could see that it was still related to Exchange, which was constantly sending me calendar notifications for the same meeting over and over. I had Exchange set up for "push', which is the default. I switched it to "fetch every 15 minutes", which helped quite a bit. However, not a long term acceptable solution for me, since I get a lot of emails that I need to respond to rapidly at my company.
So yesterday I bit the bullet and did a "restore as new" followed by "restore from backup". During this process, I noticed that an "update from your wireless carrier" (Verizon) was installed. Not sure if that is a recent release related to this problem or not.
I set Exchange back to "push". The phone held charge overnight, and was still at 100% in the morning. I was a little shocked. Will have to wait to see if the problem resurfaces, but so far so good.
Something was bugging me. My buddy at work has the same phone, and the same settings as me (push), and hooked into the same Exchange server, and wasn't seeing the issue at all after updating to 6.1. Also, my ipad was not experiencing problems, it was only my iphone 5. Maybe something went wrong during the original ios 6.1 update. I know my iphone was not connected to a power source at that time (but it had a full battery), but my ipad was connected to the charger when it updated. Maybe that was it, although the phone seemed to have plenty of charge left after the update.
It is a bit of a hassle having to reinstall all the apps, etc. But there is hope, at least. I'll update the thread if my hopes are dashed yet again.