Try this on for size:
Annual Oil Bill * Efficiency = Cost of Oil actually used for heating. (The rest is waste).
So on an annual bill of $1000, you are currently paying $650 for heat, and $350 out the chimney. (i.e. $650 is as low as you can go with a 100% furnace)
So, if you were to replace that with an 87.2% furnace, your annual cost would be $650/87.1% = $747, and there is about $100 waste. (Note! this is a savings of $250 per $1000 of Oil)
On the 86.1%, that's $650/86.1% = $755 for a difference of $8 per thousand of oil currently used. At that rate, the new furnace would take 375 * (time in which you use $1000 of oil) to pay off the difference.
This analysis is based purely on BTUs produced at the burner.
Now, there are other factors, such as air flow and distribution which may significantly affect your oil use, and the bigger furnace may be more efficient in other ways.
This is a complex problem, and it's going to be difficult to provide an accurate answer without inspecting the equipment. So this is going to be more like a few comments, than an actual answer.
Water heater
I'll start with the water heater.
Old units aren't efficient
If the unit is older, it's likely not performing at peek efficiency, which is wasting energy. Updating the unit could save you money in the long run.
Tank heaters waste energy
No matter how much hot water you use, the heater is turning on from time to time to heat the water in the tank. The only way to eliminate this waste, is to install an on-demand unit.
Fuel source cost
Depending on the costs of different fuels in your area, switching fuel source could save money. Installing a gas fired on-demand water heater, could save you money long term.
Fireplace
If the fireplace doesn't have a fan, that is blowing heat into the home. You're quite possibly drawing more cold air into the home, than the amount of air the fireplace is heating. By using the fireplace, you could be blowing money out the chimney.
Also, the cost of wood is typically quite high, compared to other fuel sources.
Heat pump
Heat pumps use a lot of electricity, which in many parts of the country is an expensive fuel source. In temperate climates, heat pumps make sense because they're not typically used much. In climates where the temperatures often drop below freezing, heat pumps are virtually unheard of. This is because heat pumps aren't so good at heating homes, and often have to rely on supplementary electric heating elements.
Use the furnace on cold days
Since you have a gas furnace in the home, you can use that when the heat pump can't keep up. Depending on the local gas price, this could save you some money. In most dual fuel systems (like yours), if the heat pump can't keep up it shuts down and the furnace runs instead.
From your description, it sounds like the furnace and heat pump are running at the same time. This is not a good idea, and can actually cause damage to the system.
You'll want to configure the system (or have it configured) so that when the thermostat switches to auxiliary heat, the heat pump shuts down and the furnace kicks in.
Without actually being at your home, it's difficult to diagnose the problem. You might want to have an HVAC technician come out and evaluate the system, to help determine if you're wasting money.
Best Answer
I think some of the answers here are confusing because they don't really address that your furnace has two different air intake/outlets.
One is for the combustion system. Air is pulled in from somewhere (usually outside, probably the metal duct on the right), mixed with natural gas, and burned. All of the outside air and combustion products go out of the chimney to somewhere definitely outside. Very hot air from the burning of gas heats a metal piece called the heat exchanger.
The "makeup air" mentioned elsewhere would be if the combustion loop sucks air from inside the house instead of through a duct from the outside. In this case some air from the house is blown out the chimney and is replaced by air seeping in through the cracks. This would not be code compliant where I live, for example.
The other air loop is through the ducts in the house. Typically a furnace will suck air from inside of the house (probably the vent behind your couch), pass it over the heat exchanger, and blow the now hot air out of the vents in your rooms. This typically does not pull a significant amount of air from outside because that would waste a lot of $$$ by displacing heated air to outside. Houses are not airtight though.
Your furnace is obviously gas because of the supply pipe on the lower right, and the presence of a chimney on the top of the unit.
There is really no way for anyone here to tell you whether it is safe or not. Age and presence of recalls are good places to start though. Adding a CO detector and making sure you have enough smoke detectors would help significantly.
If this is a matter of LIFE AND DEATH, there is not enough information in your post to provide an absolute answer. YOUR SYSTEM MAY NOT BE NORMAL.
Have you ever changed the furnace air filter?