A couple of years ago I bought an old house that was built in 1880. Right after moving in we had the old electrical fuse system replaced with a new circuit breaker box by an electrician. Our electric bills seem to be very high - almost twice that of other people in the area with similar houses. I know a decent amount about electric but am no expert. I have not been able to determine what is using all of the power. I did however notice that when the new breaker box was installed, he used all 20 amp breakers even though most of the outlets in the house are wired with 14g wire. I've read that that is not safe to do, but could that also be the cause of our high electricity usage?
Did the "electrician" pull a building permit prior to doing the work you commissioned? If not, then you have more than one problem here. If he did, presumably it was inspected and signed-off by the building inspector and perhaps you are misinterpreting what is actually installed. No professional electrician would install 20A breakers on a 14g circuit. Doing so is not only dangerous but illegal and unethical.
I suggest you contact this electrician and ask him to explain the work that was done.
If indeed the work was done by an unlicensed person without a permit, you should get a qualified electrician in immediately to inspect and correct the issues. Yes that might be expensive but a fire is going to be much more expensive.
But to address your root question, NO, this would not cause excess electricity usage. An electrician should be able to help you find the cause there as well but I'd be looking at the usual high-powered culprits, space heating, water heating, cooking.
The answer is no, the wire gauge will not affect the power usage. With that said any circuit that has 14 awg wire should be on a 15 amp circuit whether or not permitted.
Having 20 amp breakers on #14 wire is a recipe for a fire, especially if backstab connectors were used. They have enough problems on 15 amp circuits.
First and foremost, 14g wire on 20A breakers is absolutely a fire hazard, against code, and the electrician should have their license revoked (assuming they carry one). No, this configuration would not cause higher electricity usage.
A high electricity bill can be caused by high power usage, obviously. If you unplug everything then does your electric meter keep ticking? Do you have a pool heater, air conditioner, electric stove, hot tub, electric water heater, etc... that requires a constant power draw?
What might be causing your high watt/hours usage may be a side effect of bad wiring or old appliances. Is the water heater, clothes drier, or washing machine from the 60's and built to last forever?  Well, utility prices have gone up. :-( The first thing to look at when you see high electrical usage are your utilities, windows, and insulation; Old washers and driers do not work as efficiently, leaky windows and poor insulation will make heating and cooling systems run more often, if you put your electrical usage in a spreadsheet and you see a big spike with seasonal changes that's a sign that you're using your heating and cooling too much. Also, if your old house was built in the 1880's it must have a fireplace, does that have a working flue that you keep closed when not using the fireplace? [/edit]
The 14ga. wire is a definite hazard and a sign of the installer cutting corners. When I was working as an electrician's assistant I was always against that choice and argued with owners and architects that it was unsafe. It's such a minor cost-saving it's just ridiculous. But it's just a signal that someone was cutting corners. I'd expect that there were other issues in electrical wiring that were either not addressed and possibly concealed. Old water heaters can be maintained by cleaning the heating elements which makes them as efficient as when they were new, but the cost effectiveness is debatable.
Another sneaky fault can be a ground loop. The symptom might a mild electrical shock when you touch the sink faucet and the screw on the switch or outlet cover at the same time. Or if you are in the shower and touch the shower handle and the sink drain at the same time. This happened in one old house I worked on in 2012; The shower fan had been wired with 2 conductor lighting cord which was tied to both the ground and the negative side of the fan so when it was on there was approximately 70v on the ground all around the house.  A ground fault can significantly increase the current draw of a fan but the only sign would be an annoying low current shock when touching the drain and the handle of that shower. That type of ground fault can raise the cost slightly, but it's a hazard as well. [/edit]
After 8 years as a full-time assistant electrician in the 90's my experience supports everything everyone has said about fire hazards and the 14ga. wiring. I would never put that stuff in a house. Maybe an LED lighting system in an RV. However, that's not going to cause high watt/hours on your electrical bill. [/edit]
Recently helped a friend with a wiring issue in their basement. Circuit was blowing fuses. House was built in the early 70's here in Virginia. When they put in basement wiring with a 14ga. branch to an overhead light on the workbench, they forgot to think about the obvious. Somebody down the line will put an outlet on that circuit and plug in a dehumidifier that runs at 11-12 amps all the time. This caused a higher current on the smaller gauge and that was focused in the junction box where the 14ga. branch was tied in to power outlet. after four decades the outlet burned out and part of the copper was touching the ground. Put a fuse in, it pops (fun!).  The point is, 14ga. wiring is dangerous if it is misused, but that alone doesn't cause high current draw. [/edit]