That is an oddball setup. In a closet and no return ducting at all? Yeah I don't know if there is much that can be done. You may be able to reduce the speed of the blower motor, this can change the entire heating/cooling profile of the unit though. I would probably get an hvac tech to help with that just to make sure you won't cause issues. For A/C mode you probably have no choice but to run it at full speed as you don't want the coils to freeze up.
I would try to find a way to redirect the return vent to the basement, that may help a little bit. Provided there is an air route to the upstairs. Ex: if the basement has a door to go down, make sure that door has louvers, or that there is a vent somewhere else in the floor. you don't want to be sucking air from one closed space (basement) and trying to put air in another (upstairs) but if they are open to each other then it's ok. Once that is setup, and add padding to the entire closet. Make sure it's something fire rated just to be safe. Roxul insulation might work, provided you do in fact redirect the return somewhere else. That stuff can create dust so you would not want to leave the return in there. Filter would probably take care of that though...
If you do end up finding a way to redirect the return out of that room, you could also replace the closet door with an outside door. They block sound pretty decently.
Also it looks like there is only one PVC pipe unless it's off the side and just not seen. If yes, consider locating where the combustion intake is, and add another pipe going outside. Not having both kind of breaks the efficiency rating because it's using indoor air (that you are paying to heat) for combustion then sending it outside. That won't help much for sound though.
Another thing that may help, although pretty hard at this point, would be to jack the furnace up and lay it on some kind of rubber matting. Normally furnaces are in the basement sitting on concrete and not on an upstairs floor so I imagine the various vibrations of it are resonating throughout the whole floor. The gas line, duct work etc should have a little bit of play in it to let you do that, but I'd maybe do it as last resort. Basically do all the other fixes first and see if those are good enough.
Edit: Looking at it again I just realized that bottom part actually IS ductwork. What is immediately under it? You could cut a hole in the floor and then go to the basement that way. Just don't cut the joists, it's ok if they are in the middle of the air path, return vents are sometimes even built right between joists so you could even cut the hole, then cut a hole elsewhere in the basement between the same joists to take the air from there if you don't want to have the vent in that location, or can't.
