The main issue that I can see is the Microsoft patents that loom over its head like a sword of Damocles. Microsoft has made a "patent promise" not to sue Mono users for infringing their patents, but that's still a bit... unsettling.
You'll find quite a few Linux and free software users very wary of a Microsoft technology with such patents; the open source world has a peculiar relationship to Microsoft, and though they seem to be more friendly to open source since a couple of years, the "embrace and destroy" strategy is still a threat. Whether this fear is justified or not, I can't judge - but I understand the caution.
As for Godot, as long as we're not made to be dependent on Mono, I don't see any issue. Best case, everything goes well and we're happy with our Mono version; worst case, we drop the module and continue with the other languages we already have and intend to keep.
Also as a Linux packager, I can tell that (at least the open source ones) Linux games (and other applications) relying on Mono are a real pain to package and maintain. In my experience the Mono stack is as unfriendly to packagers as Java; that doesn't make it a bad technology per se, just not one I want to fiddle with too much :P