My guess is a short in the wiring. Here's some information I found online, perhaps it will help you troubleshoot and identify the problem:
The parts of a typical doorbell system are the push button, the doorbell, chimes or buzzer and the transformer. The transformer allows the bell to operate on low voltage (6 to 24 volts.) Wired into the circuit at a junction box, it steps down the voltage from the regular 120-volt circuit. When your doorbell doesn't ring, or rings constantly, the problem may lie in one of the parts or in the wires that connect them. NOTE: To diagnose most problems, you'll need to have the power source connected. But if you're going to work on the transformer or the wires in the junction box, be sure to shut off the power to the circuit. Remember that the input side of the transformer is high voltage, 120 volts.
When the Doorbell is Silent:
The first place to look is the source of power. Make sure a fuse or circuit breaker hasn't blown or tripped. Once you're certain that the 120-volt side of the transformer is getting power, shut off the power and tighten all wire connections. Turn the power back on and check the low-voltage side, following the steps below to find the source of the trouble.
Testing the Transformer:
1) Use a volt-ohm meter. If the transformer is working correctly, the meter reading should match the secondary voltage (6 to 24 volts) marked on the transformer or bell. 2) Set the voltage range on the meter to 120 volts AC and measure the voltage between the two low-voltage terminals on the transformer. 3) If the meter reads significantly higher than the correct secondary voltage, the transformer is defective and should be replaced. 4) If the reading is close to the correct secondary voltage, test again by setting the voltage range on the meter to a lower value. If the new reading doesn't agree with the voltage marked on the transformer or bell, replace the transformer.
Testing the Push Button:
1) Disconnect the two wires connected to the button and short them by touching their bare ends together. 2) If this makes the bell ring, the push button is defective and should be replaced. 3) If the bell doesn't ring, the problem is either in the bell or chime mechanism or in the wiring.
Testing the Bell or Chime Mechanism:
1) Have a helper push the doorbell button while you listen to the bell. If it makes a buzzing or humming noise, it may be gummed up with dirt. 2) Check the mechanism and clean it as necessary using fine-grade sandpaper to remove corrosion form any contacts. 3) If the bell still hums or buzzes after cleaning, replace it. 4) If the bell doesn't make any noise at all when the button is pushed, disconnect the bell and, using new wire, hook it up directly to the transformer. If it works, inspect the old wiring. If it doesn't, replace it.
Repairing the Wiring:
1) Examine the wiring for breaks or frayed insulation that may be causing the wires to short out. 2) Repair any breaks and wrap the repairs with electrician's tape.
A Constantly Ringing Doorbell:
If a doorbell rings constantly, either the button is stuck or the wires going to the button are shorted together. To test follow these steps:
1) Test the button by turning off the power to the transformer. 2) Remove the button from the door frame and disconnect one of the two wires connected to it. 3) Turn the power back on. If the bell doesn't ring, the button should be replaced. If the bell rings, the problem is a short between the two wires. 4) With the power turned off, examine the wires for frayed insulation or bare wires rubbing together. 5) Use electrician's tape to wrap them where necessary. 6) If you can't find the short, replace the wires.