Those louvered bifold doors would be a big part of it. They'd stop next to no noise at all.

If it wuz me I would go to your local Habitat ReStore and buy a few sheets of plywood that you can cut to the size of opening to cover the bifold doors. Glue/screw them together and sit that mass of wood in the opening in front of the bifold doors. See if that helps. You can also replace the drywall around the closet with Dow Corning's QuietZone drywall. There are other drywalls that reduce noise, often by simply being heavier than normal drywall, since it's primarily mass that's effective in minimizing noise transmission through walls, floors and ceilings.

Or, at least, I'd focus on containing the noise within the closet as best you can before you start replacing the furnace cuz of the noise.