I know some people do mountain bike on fixed but I think they are insane highly skilled riders who crash a lot are very good at reading the trail and not digging pedals into ruts and the like. I think gearing would be a big problem anything low enough to get you up the hill will be seriously spinney on the way down.
If by off road you mean a gravel cycle track or two....why not.
Tyres are no problem I bought some knobblies for my wife's hybrid earlier in the year, just make sure you have clearance for the tyre.