This happens when the contacts between the screen connector and logic board are not seated correctly. When you first switched it on and you saw the "rainbow" screen; this is a tell-tale sign that the screen is not connected right.
Inspect the connectors on both the logic board and the screen. The connector for the screen is shown in ''step 30'' of this guide: [guide|3148|page=10]
I'm not saying you were careless but you have possibly damaged a connector pin, i did this once on an iphone3GS and experienced the same problem.
The reason it is getting hot is that there is a short or a loose connection between the two and causing the components to over-work inducing heat, in some cases they may have even become damaged.