Your question is too wide I am not sure a clear answer can be given but I would try. As such sitting from here I cannot tell you what could break after chaning compatibility level. Only you can find it out by running each and evry procedures,functions, views and triggers you have.
Microsoft has documented breaking changes in SQl server 2008 R2 before changing the compatibility level you must go through this article.
You must also read Change database compatibility level BOL document to see what could get affected after changing compatibility level. Please read complete article most of the things are documented.
Did you ran upgrade advisor before you migrated database to 2008 R2 ? that should as well have pointed out breaking changes.
The good thing with compatibility level is it can be immediately changed to previous value if you face issue. So you also have got hit and try method to check out