I have a FLEX application created with the 2.2 ESRI Library. Will my application break if I transfer it to ArcGIS Server 10.1?
Probably not.
The only forward-compatibility issue in the Flex API I'm aware of is using 1.x to connect to a tiled map service that is from a 10.1 service, but if you are using 2.2 then you should be fine. However, this is not a use case we have tested extensively.