Is there any way to stop the air leak through my furnace without compromising its operation?
Judging from the previous picture, I thought it returned from the lower floor through the bottom of the furnace. If it looks just like that new pic; a side tap with plenty of room, you could put a low-leak damper there that the blower will pull open all by itself when it runs (I'm having trouble finding the kind with a small weight attached...). This will block most of the flow in the ductwork, which is exchanging heat and causing a draft (half of the problem) where it's run in unconditioned space.

How is the ductwork insulated that I assume runs through the attic?  Your plenum chamber should have self-stick insulation on it, if it's in an unconditioned space as should any ductwork.
There is also a noticeable lack of sealant on the system allowing unconditioned air into it (the other half of the problem). My new best friend is duct sealant mastic in a bucket; tape needs the proper surface prep and can does eventually fail.
Steps one and two for addressing rigid ductwork in unconditioned space:

Another possible problem is that you have not truly sealed the mechanical room from the rest of the house including, however small, the return area.

Things to consider:

Installing a new system may require a completely different plenum; you should seal it for now but maybe wait on insulating it.

Jacket loss on equipment is horrendous, even on a well sealed and buttoned-up install; I prefer it in conditioned space. It's more efficient and less considerations need be taken: freezing temperatures; insulation is necessary.

You've used the word pascals, which means you know more about it than I do, however I'd of been quite satisfied blocking the air intake and installing a louvered door to include the room into the 'house'. If it is completely spray foamed (hermetically sealed) I'd be concerned, otherwise...meh. (Provided return air is not taken from the mechanical room.)



702.1 All air from indoors.
Combustion and dilution air shall be permitted to be obtained entirely from the indoors in buildings that are not of unusually tight construction. In buildings of unusually tight construction, combustion air shall be obtained from the outdoors in accordance with Section 703, 705, 706 or 707. -iccsafe.org


This intake will disappear (along with most of the problem) once you put in a HE system. Decide when and what you're going to spend money on. Recommended order: 1. duct sealant 2. HE furnace 3. spray foam insulation.

Having to install a damper is not a common solution. Attack the root of the problem; air infiltration and heat exchange.