I got an iPhone 5s on my birthday in June, and had the same overheating problem that other people had - I browsed through this forum looking for some solutions and in the process found that the 'battery and usage' section had a bit to offer in terms of my issue (Setting>General>Usage>Battery Usage)
In that section, I noticed that there was a breakdown for the battery usage of all my apps - which I believe is a feature in the most recent iOS version, and saw that the background activity for Facebook was responsible for 95% of my last 24 hours of battery usage! Whilst the app background refresh option was enabled, I found that my iPhone would overheat, and the battery would deplete very quickly, but since disabling the app refresh, and a few other things found here: 'http://www.zdnet.com/pictures/improve-iphone-ipad-battery-life-ios-9/23/' I have had my iPhone on and idle for about 45 mins, and the battery has only gone down 2% whereas before doing those things, it was draining roughly 1% every couple of minutes or so.
If this helps anyone at all, you're more than welcome - if not then I can only suggest that you try something else