What are the typical reasons for bugs and abnormal program behavior that manifest themselves only in release compilation mode but which do not occur when in debug mode?
It can, especially if you are in the C realm.
One cause could be that the DEBUG version may add code to check for stray pointers and somehow protect your code from crashing (or behave incorrectly). If this is the case you should carefully check warnings and other messages you get from your compiler.
Another cause could be optimization (which is normally on for release versions and off for debug). The code and data layout may have been optimized and while your debugging program just was, for example, accessing unused memory, the release version is now trying to access reserved memory or even pointing to code!
EDIT: I see other mentioned it: of course you might have entire code sections that are conditionally excluded if not compiling in DEBUG mode. If that's the case, I hope that is really debugging code and not something vital for the correctness of the program itself!