The symptoms would indicate two issues. The relay would click and energize when the horn button is depressed. But, when the relay energizes, the horn (or horns) should sound. If it stayed energized, it would heat up and draw current. Strange but both the horn (horns) would need to be bad and there is a short in the horn buttons or wires leading to the relay. If just the buttons are shorted, the horn should sound constantly. If just the horn is defective, the relay would not click unless the buttons were pushed.

Or a third possibility; could the three wires to the relay on the wrong terminal? If the horn button and horn connection were switched the relay would be activated all the time but at a reduced current from the resistance of the both the relay coil AND the horn coil. Thus, the relay would be on all the time, the horn would not have sufficient voltage to sound, the relay would heat up, and the battery drain. It is a quick thing to check anyway.
see: http://www.thefirstgensite.com/library/electrical/67wir/67wir2.html