Doorbells should have about 24 volts AC across the button when you're not pressing it. Here's what you do. First look at the breaker panel and make sure all the breakers are on.

Now look around the basement for something that looks like this. (good places to start are right near the panel but you may find it just nailed to a floor joist or something):

That's the door bell transformer, again there should be 24VAC on the screws on it. If not, trace the power wiring (to find out if it is being fed with 120VAC).

If it is then follow the wires looking for damage.

It is certainly possible that the electrician or the installer messed something up. Are you paying both of them directly or is the furnace contractor paying the electrician? The electrician might have unhooked the transformer in a panel or junction box. Someone might have accidentally cut the low voltage cable, the button could be broken or the doorbell could be broken.

Do you know where the doorbell transformer is? If this is a typical installation, it makes 16VAC from 120VAC. The transformers are frequently mounted directly to the main electrical panel or a sub panel, but they can be anywhere on a junction box as well. I have seen them on a combined circuit providing power for a furnace or boiler. If you can find the transformer, it might be easy to tell if it still has power because it will be a little warmer than the surroundings and it might be humming a little. (I can't say what tune, they all have their own preferences.) The easiest way to check to see if the transformer is producing 16V is with a test light. You can rig that will a robust automotive lamp and a couple of wires (preferably with alligator clips on the ends. See if the test light gets bright by touching the wires to the low voltage output of the transformer. If that checks ok, you have to test the doorbell, the button and trace the circuit to see if it has been cut.

The circuit is simple and uses low-voltage, two conductor, cable. There is a loop between the transformer and doorbell with the button on one conductor in there somewhere. If the circuit is intact, you should have power at the doorbell when the button is pushed.

The electrical company was a subcontractor of the furnace company, so I paid all funds to the furnace company. I bought a new doorbell button and installed it to see if that would solve the problem, but it didn't. I'll look for the gizmo that Ron showed me. As for the test light, I think I may have one buried in my tool box somewhere.

Substituting components is not a bad approach in this case. You can use a meter to test output, but electronic versions can trick you with induced currents. Looking for low voltage is even worse than line voltage in that respect because you can pick up static from adjacent 120 or 240V cables.

You can probably fix this or, at least, pin it on the installer.

Thanks again, guys, for your help. I emailed the furnace contractor and raised the possibility of the furnace installation being connected to my doorbell problem. The operations manager was all over it, saying that it's not at all uncommon for there to be a problem because the furnace and doorbell work off of the same transformer. He sent a guy the next day. The guy showed me the transformer, just like in Ron Natalie's photo. It was right next to the furnace. The guy messed with some wires and got the doorbell working again.