On Mac OSX, it's pretty straightforward. Multiple applications can access a midi controller at the same time. Both Resolume and the other app can have midi input enabled, both will listen and respond. You might just want to make sure only one sends back feedback.
On a PC, things are a bit more difficult. Apps cannot share a controller, so you will have to use a midi routing application like Bomes. You can create virtual midi outputs in Bomes, and you can forward incoming midi data to multiple virtual outputs. So your controller would send midi data to Bomes, and Bomes will forward this to two virtual outputs. Resolume will then listen to one virtual output, and the other app to the other.