When I flew PA34 for living it was quite common to for the ASI to drop to 0 in icing conditions - even though the pitot heat was on. Once we were shown this it was a non event. We just continued to fly and after a while the indications were coming back.
Having said that, one colleague of mine, when faced with this situation for the first time, reacted on instincts and pushed the yoke full forward, sending the plane into a dive. Luckily they recovered before the ground...