I would and have done. If he's wrapped up he'll be fine, and it's not like you have a lot of choice if the dog needs walking. Babies are quite robust, mine have survived being dragged out twice a day in all weathers.
Definitely get out. A change is as good as a rest and all that. Baby might be distracted by being outside and if teeth are still bothersome later you'll feel less frazzled coping for having had some good fresh air rather than being cooped up all day
I second JugsMcGee suggestion, wrapped snug in a sling, he'll find it comforting to be close to you, you get a lovely baby hot water bottle attachment, dog gets a walk and Ds gets a change of scenery to take his mind off his poorly, irritated gums. Have a lovely walk
It's -2 outside (but gloriously sunny) and I've got a very clingy tearful teething baby, for whom I've already decided not to go into college, so that I can bf on demand today. However, I do need to walk the dog - it's sooo cold outside, but would it still be ok to wrap DS up all cosy and bundle them both out to the park?
I have to say if I was feeling like DS is, it's the last thing I'd want.