Not in the sense we're discussing it here, they don't.

There's a list of about 20 games said to have DRM in Gog and when you actually read the list rather than just it's title it turns out none of them has what we would call DRM - any sort of phone-home validation or anti-piracy measure.

It's mainly things games with add-on content that requires you use Gog Galaxy or register online, some that send analytics to a server and stuff like that.

You can see the info here,

Whilst it's still nasty and still shouldn't be happening, none of that makes the game unusable in the future after the servers are down if you still have the offline installer.