Recently we got some (natural) gas logs which work very well, but I can tell
that probably more than half the heat goes up the chimney.

I remember years ago when we had unvented gas heaters in every room. There
seemed to be no danger then.

My question is, is there any danger to plugging up the chimney, at least
partially, so that we get some of that lost heat in the room?

I believe the danger is not carbon monoxide, but oxygen depletion. This is a
very old house and drafts alone should be enough to prevent serious
depletion.