In an application currently in production, one module of code is being modified. Is it necessary to re-test the whole application or is it enough to just test functionality associated with that module?

Actually it depends on the module you are modifying. If this module have no impact on other modules, you don't have to re-test entire application. If the change in your module have functionality association with other modules, you need to test all those modules as well.

You need to re-test (need to do atleast sanity test) the whole application, because you may not know some times, the one which you have modified is going to effect for other modules in the application.