There are a few things in DisplayFusion that require an internet connection:
- Update checks (this can be disabled)
- Online wallpaper (Flickr, Vlad Studio - both are optional and not required)
- Certificate checking

Sometimes Windows will check the signing certificate that is used to make sure it is valid, and the program hasn't been compromised. This check is not performed by DisplayFusion, and doesn't connect to any DisplayFusion servers. This is a Microsoft thing, and something that can't be disabled (as far as I know).