How to insulate around a gas fireplace without tearing it apart

I assume your gap is on the indoors side?

Regardless, if air is filtering through into the living space, there must be a gap outdoors ( external chimney) as well as indoors.

I recommend you check out the caulking along the chimney to the exterior siding as well as addressing the interior gap. Since this gap inside is directly adjacent to wood trim, I also have to assume that excessive heat is not a factor. Is the metal you are referring to an insert trim of some type? A pic would be a big help here.

If the assumption are right and the gap is 3/4 inch wide, I would use a caulking backer in the gap before using a good grade painter's or acrylic caulk as a topper. The caulking backer is a round foam solid tube that can be cut to length, inserted into the gap to seal it and hold the caulk. After the caulk has cured, it can be painted to match the trim. If for some reason you are concerned about high heat in this area, there is a fire rated caulk made especially for gaps in fire or common walls. Readily available at any box or hardware store from a number of manufactures.

Ok after reading your comments, and learning that the black vertical sides are slate, I would still do similar to above. Perhaps after injecting larger amounts of fire rated caulk as deep into void as possible, then use the solid, tube style caulking backer to fill the majority of the larger, deeper gaps, followed by using some paint-able silicon/acrylic mix caulk.

Before you apply the finish caulk, run a nice straight line of painter's tape up the front of the slate even with the edge of the wood trim. When you fill over the foam backer with the caulk, use a small putty knife to draw the caulk smooth and flat. Immediately remove the tape while caulk is fresh. This should result in a nice crisp straight line. Let the caulk cure for 24 hours. If there is some shrinking or cracks, no problem, simply repeat the process with new tape and another coat of caulk.

Alternately, you could also use a setting type drywall mud, such as durabond 30 or 45 to fill the deeper gaps etc. Then again use a second coat to create a smooth straight extension of the wood trim to the slate.

Either method will result in a neat, easy to paint fill that should stop your air infiltration problem and make the finish look nice.

It is very important to ensure all infiltration paths a sealed off and adjacent walls and floor are properly insulated. Even a hairline crack in the doors can cause enough infiltration to lower temperatures in the area, though no actual moving air can be detected.

Even so, the fact that outside air can freely circulate in the combustion chamber will also be a heat sink, lowering temperatures in the area. More sophisticated devices similar to this utilize automatic dampers that interlock with the fuel supply. These are located on the exhaust flue, and sometimes on the combustion air inlet as well. I don't know if retrofitting such devices is possible on typical residential fireplaces. Such a project must be done very carefully to ensure complete safety. Failure of the system can result in carbon monoxide poisoning, so multiple fail safe measures are called for. Such measures are beyond the abilities of nearly all DIY practitioners. Only fully qualified professionals should perform this sort of work.

Even more so, though flue dampers prevent cold air from freely circulating, they are typically thin sheets of metal with poorly sealed edges, so the fire box will still be fairly cold, albeit much less so.