An apparatus (1) for mounting and removing a tire (2) on and from a respective rim (3) comprises: a movement element (9) for moving a bead (2a) of the tire (2) in order to insert/remove the bead (2a) into/from a respective groove in the rim (3); means (4) for supporting and moving the rim (3), designed to rotationally drive the rim about its longitudinal axis; characterized in that the means (4) for supporting and moving the rim (3) comprise a sensor (252) for measuring the mechanical strain applied to the supporting and movement means (4) as a result of the mechanical stress on the bead (2a) of the tire (2) during mounting/removal of the latter on/from the rim (3).