Furnace next to your room, right? With exhaust pipe from the burner coming up from it, venting out the roof?
How much space under the door? Enough for some water tubes/pipes? Run that heat exchanger from PedalingBiped from it. Find a small heater core, and insert it between two sections of the exhaust pipe. It will probably not all fit within the pipe, so you will want to enclose it sufficiently to ensure no exhaust leakage from it. You may also need to clean it regularly to prevent buildup on it that could block the rising heated exhaust from exiting the house. I'd actually do it inside a new section of exhaust tube that is larger diameter than the existing stuff, joined properly to the smaller stuff above and below it, completely enclosing the core.
Use thick plastic tubing and insulate it, for the part coming from the core in the furnace exhaust tube to the one in your room, so that as much of the heat as possible makes it from the furnace core to your room's core.
If the water in the core is heated sufficiently, and the diameters of the pipes to and from it are right, with one-way valves, you may not even need a pump, depending on height difference; it could be forced down by gravity and back up by pressure created by the heat. Probably you would need a pump, though, perhaps a simple PC-cooling system pump would be enough.
If it is left running all the time, you can use a pretty small system, perhaps even using the PC-cooling system cores, too, and the low-volume quiet fan.
It's still inefficient, but the only extra power you are using up is that of the pump and fan, which is fairly minimal. The rest would have been waste heat from the furnace, anyway.
It may not be safe to block the exhaust riser of the furnace if it burns anything other than natural gas/propane/etc, as soot could quickly build up due to velocity drop of the exhaust, and clog things up. That's something you'll want to double check before trying it.
Regarding a water tank to redistribute heat from daytime into nighttime, it's perfectly feasible, assuming you have the tank itself within the space you want to heat. Otherwise you must move the water and/or heat around a lot more, and insulate the tank and distribution system quite well (especially if the tank is to be outdoors). If it's in teh same room, it needs little or no insulation. Just pump the water on top of the tank out to the solar heaters, then back into the tank along the bottom (because warm water sinks anyway, and the less stirring of the tank you do, the more cold water you reheat, rather than having less energy differential with already warm water).