You can build a circuit to do what you're asking, though as others have pointed out it's not necessarily the best way.
What you can do is use a 24V AC relay to switch an outlet.
Waring: 120V AC is dangerous and can seriously hurt or kill you. If you're not aware of what the difference between hot, neutral and ground is, or the thought of "exposed terminals" doesn't scare you, stop now. Learn some basic electrical skills, find another way, or enlist the help of a friend.
Get a relay
You need a relay with a 24V AC coil, and which is rated to switch 120V AC (or higher) at 15A (or higher), it just needs to be SPST (single pole, single throw) though DPST or DPDT (double pole, double throw) is fine.
Mount the relay in a box
You need a box to put the relay in to protect it from being touched (since there will be 120V on the terminals of this thing). There are many types of boxes, from metal junction boxes to PVC project boxes.
Figure out some way to secure the relay base down in the box (PVC is a lot easier for this).
You'll power this box by plugging it into a regular outlet. You need a power cord and a strain-relief connector.
Get a receptacle
Probably the easiest way to do this is to use the female end of an extension cord and a strain-relief connector. You can actually just buy an extension cord and cut it and half, and use this for the power in and out. You can even use one of the fancy ones with the lighted end, and you'll have an indicator if the relay is on or not.
Otherwise you need a normal receptacle which can either be mounted in this box, or attached via a surface-mount box (aka handy box). If you attach it externally, you'll need to use either armored cable (BX), or extension cord wire (SJO) and appropriate box connectors or strain relief connectors.
Wiring it all up
Here's the wiring diagram for your furnace:
The relay you get will have a pinout on it, something like this:
- Connect your furnace W and C terminals to the coil of the relay (terminals 2 and 7)
- Connect the hot (black) wire from the 120V plug to the power terminal of the relay (terminal 1)
- Connect the hot (black) wire of the receptacle to the switched terminal (3)
- Connect the two neutrals (white) together.
- If you have a DPST relay like this diagram is for, you can optionally switch the white wires by connecting to terminals 8 and 6, mirroring the hot wires. It doesn't really matter.
- Connect all the grounds together, and if you're using metal boxes, the grounds must connect to the ground screw on the box itself.
That's really all there is to it. When the thermostat calls for heat (powers W terminal), 24VAC will power the coil of the relay, which will turn the receptacle on.
You will likely be able to find all of this at your local box store. If they don't have relays, a local trade electrical or industrial supply store likely will, and I hear this internet thing sells stuff now too.