I'm having the same problem, but it's only when I am pulling data as you said. If I put on airplane mode, I can play games, listen to music, and use other functions with a normal battery drain. As soon as I start pulling data, whether I'm surfing the web or talking on the phone, it heats up and drains the battery. After talking for nearly 15 minutes, it will go to the "overheated" screen and won't let me use until it's cooled down. It heats up mostly around the SIM card side of the phone. I've been reading forums for quite some time and can't find any solutions to this issue. I've shut down all recommended outside apps, but to no prevail. (Also, what's the point of having an Iphone if I can't use apps, location services, siri, email, etc..)
If you have any answers, please let me know.