Replace what you have with similar is usually logical. Modern laundry usually has a vent, so at least a gap at the bottom of the door is necessary. Furnace room will have more important code requirements. Old furaces and water heaters drew air from the room. That’s no longer allowed in new appliances. So if they do draw from the room, you must have ventilation. If not, it doesn’t matter / isn’t required. Either way, that doesn’t prevent you from installing solid doors. Also, you can vent through the interior wall if you don’t want to cut into your door.
Not all modern furnaces and water heaters pull in outside air. Less expensive gas furnaces may still use interior air for combustion and almost all gas water heaters will. Heat pump water heaters absolutely need ventilation to work correctly as well.
Find the studs, block out a rectangle of necessary size to mount vent grills on both sides. Essentially, make your own hole in the wall and add 2x4 to the side(s), top, and bottom. If you plan it right, you won’t need to cut open the wall more than the finished hole size.