Do pilots adjust the aircraft's flight path to allow for the curvature of the Earth? If an airliner is flying at approximately 500-600mph, it seems to me that there would need to be a significant adjustment for altitude so as to not fly off into space.

Wikipedia says that there's an 8 inch drop for every mile because of the Earth's curvature. But, I've never heard of any airliner adjusting for the curvature. Also, shouldn't aircraft have to adjust somehow for the Earth's rotation because it varies depending on the latitude?
Aircraft altitude is measured (inferred) by atmospheric pressure. The aircraft is usually flown at an altitude that maintains constant ambient pressure (by pilot or autopilot, as the case may be). Changes in local barometric pressure (provided by air traffic control) are used to recalibrate the aircraft altimeter.  As long as the aircraft is flown at a constant ambient pressure (hence constant altitude), it will be following the earth's curvature (as the atmosphere is attached to the spherical earth and has same properties at same distance from the center, in an ideal case) as the altitude is measured from the surface, which is curved, and not a plane.