This can have a disastrous effect on the aircraft, particularly in the case of helicopters, some of which need the rotor to constantly be under a non-zero amount of load.
To produce 0g, the aircraft has to follow a ballistic flight path, which is essentially an upside down parabola.
Later, it was found that these accidents usually happened during low terrain flight after passing a ridge and initiating a dive from the previous climb.
Articulated and rigid rotor systems do not lose controlling forces up to 0g, but may encounter this depending on their flapping hinge offset from the mast.
0g forces are a minimal problem for fixed wing aircraft, but there are exceptions, including, but not limited to, airplanes with gravity-fed fuel systems.