I recently had a renter ask me if it's possible to board up the fireplace, as it will not be used. SFH, wood fireplace. They would like to put some furniture in front of it. Should I have any concerns?

They can block it by putting something in front of it or a piece of plywood held in place by a piece of furniture, but I wouldn't nail boards to the  fireplace.

Is it drafty? They can cut a piece of insulation and place it under the damper. I wouldn’t let them nail boards over it either.

I don't mind doing it myself and I would actually prefer it like that. I could have a piece of plywood nailed to a wood box frame installed in the opening, to hold it in place. Paint it in the same color of the fireplace, so it should look quite neat overall. Also, by using screws, it should be pretty easy to take it down in the future.

Now I'm just thinking if there would be any legal, insurance, etc, issues with this.

Or you could just install doors, which would help with draftiness, and be an attractive permanent solution. If you seal it off completely and tell your insurance company it’s non-functional, you might even save a few bucks on insurance costs.