You normally want to remove background noise from dialogue. In this case I want to do the opposite, removing the dialogue from thebackground noise, keep the background and boost it in Adobe Audition. Does anyone know how to do it?
Depends. If it's a stereo file and the dialogue is center-mono than using the center-channel extractor will at lest reduce it. If there's any reverb on it, you'll still hear it if it was applied in stereo. If it's a mono file, then give up; you won't be able to extract the voice at all - generally noise occupies the same bandspace as speech, so there's no way to differentiate them.
What about boosting the background noise without affecting the dialogues?
There is conceptually no difference - it's just a matter of where the sliders are in the center-channel extractor.