Try resetting the alarm, using the instructions given at the link below. I'll also copy the text here.
- Insert the ignition/door key in the door lock on the driver's- side door of your Mountaineer.
- Turn the key to the lock position.
- Turn your key to the unlock position.
- Turn your key to the lock position.
- Turn your key to the unlock position. Your alarm is now disabled. If your key fob works, you can also press the unlock or alarm-disable button on the fob.
- Start the Mountaineer's engine and allow the vehicle to run for at least 10 minutes. This will allow the alarm to relearn the default settings. You must make sure that you have disabled the alarm first, however, before you attempt to start the vehicle, so that the alarm will reset itself.
If these instructions don't work, you can try resetting the car's computer by unplugging the battery for half an hour. However, that will reset the keyless entry code, as you can see in the fixya article I link to at the bottom