As #chad says, one or both of the leaf switches that fire the sling when the ball hits the rubber are gapped too close. The vibration caused by the flipper firing is sufficient to momentarily close the gap causing the sling to fire.
This is also likely to be the issue when the sling was machine gunning. If a coil shorts it just doesn't work - and usually causes all sorts of electronic shennanigans with the driver board.