Do termites bite?

Can they spread disease?

Yes, termites do bite people if you get them angry enough.  With most species they have to find the thin skin between your fingers (or similar) before you'll even notice.  They don't set out to bite people, but they will bite in defense.  Bigger species like Mastotermes, Macrotermes and some dampwood termites are much better at being noticed. I've seen small blood marks from a Mastotermes bite near a technician's navel.  Strangely, it is much more often the nondescript workers that bite (me) than the big-jawed, heavy-set, scary-looking soldier termites.  Alates (swarmers) may have bitten me.  I've never noticed.

As for disease, there's no available evidence to suggest that termite bites have ever transferred anything to humans.  They don't remain sensitive and never seem to swell.