What you are describing sounds very much like a piezoelectric 'buzzer' or alarm source. You can verify this by checking the underside of the board; these devices need only two working contacts to sound off. You may find a third which is sometimes used as a 'key' to insure proper orientation.
Thermal fuses are normally mounted to or near the primary source of heat and are used to prevent catastrophic failure of a high powered heating element of some kind. Their failure (blows/opens) when a design temperature is reached and if the common type, about 1/4" in diameter and ~ 5/16" long, these do not 'self-heal.' These can often be identified by a distinctive shape; they have a pointed end of epoxy fill and the other end is actually part of the case.
There is a disk or cylindrical thermal element that is a temperature-sensitive switch using a bimetal disk to interrupt current at its design point and these do reset after cooling.