How do you soundproof a door that's never going to open again?

I did almost exactly this on a dead back door in my old house and the gaps matter more than the door, at least at first. A hollow door with a half-inch daylight gap around the latch side will leak more sound than a solid door with no gaps. Sound is basicly air moving, and any open path wins. So start with the frame. Backer rod pushed into the bigger gaps, then a bead of caulk over it. Where the door meets the frame, use adhesive foam weatherstripping and compress it good when you close it. That bottom corner with daylight is probably your worst offender, so get a door sweep or a threshold fill in there. Run the hand test again after and keep hunting untill you feel no airflow anywhere.

Once it's sealed, then yes, add mass to the door itself. Hollow doors are terrible at blocking sound. The cheap route is a layer of 3/4 plywood or MDF cut to fit inside the door face, screwed and glued on, then finished however you like. Heavier panel means better blocking, simple as that. Rigid foam board alone won't do much, foam is too light, but foam glued to the door with a plywood layer over it gives you a little decoupling plus mass. If you want to go further, a second layer of drywall on top of the panel works too, it's just getting heavy at that point.

The one thing I'd check before you seal it forever: make sure nothing in that room depends on that door. Egress rules mostly apply to bedrooms needing a way out, and you've got a window presumably, but it's worth a thought if this is a bedroom. Some folks also bolt or screw the door shut first so the seals stay tight and nothing shifts. I screwed mine thru the frame with trim-head screws, filled the holes later, zero regrets.

Biggest mistake people make with this is doing the mass first and ignoring the leaks. Do the sealing, then the panel, and you'll knock the dog barking down to barely noticable.

Agreed on doing the leaks first, that tracks with what I found on our basement door. I slapped a heavy MDF panel on the inside face before I dealt with the gaps and the difference was underwhelming. Then I finally squished some foam weatherstripping down properly along the latch side and the improvement from that alone was bigger than the whole panel. One detail that helped me find the last leaks, I went around at night with a phone flashlight pressed against the outside of the frame while the room was dark, and every gap lit up like a holiday decoration. Found two spots near the hinges I never would have caught by feel.

This is helpful because I've been wondering the same thing about a door in my garage that never gets used. My worry is that even if I seal the door perfectly, the sound might just come through the wall around it, since old porch conversions sometimes have thin walls too. When you ran the hand test, could you actually feel the air coming through, or is it more of a hear-it-than-feel-it thing? I tried it on mine and couldn't realy feel anything, so now I don't know if the door is even the problem or if I'm chasing the wrong spot.