Warning: speculation from first principles:
Compressing air produces a lot of heat, so I suspect the compressor would be fine if it starts at all, which it should. If there's a lot of humidity in the line and you don't have an air fryer, there might be an issue at the tools, for the reverse reason.
And again, if there's enough humidity to cause serious build-up of water in the tank that might eventually be an issue if it freezes and can't be drained until the tank warms back up.
But in most places winter air tends to be relatively dry anyway ..