In older aircraft, loss of airspeed data didn't really do all that much to the aircraft. The autopilot might drop out, and the aircraft may give you a warning, but the coughpit isn't likely to be all that noisy a place. But, in the new ones (and I'll use Airbus simply because I'm most familiar with what they will do), there will be multiple alarms, associated with loss of autopilots, auto thrust, flight directors, reversion to alternative flight laws (which totally changes the way the aircraft behaves). The aircraft is quite likely to be out of trim, and because AB only give you pitch trim in direct law, and never give you any ability to trim in roll, you won't be able to do anything about it. Basically the aircraft will revert to a state in which it is surprisingly difficult to fly...but, and I'll emphasise the 'but', it should not be an issue to any properly prepared crew, nor should it result in any flight path deviation.