"Relativistic mass" has been a discredited concept for more than 80 years. It was one way of understanding special Relativity without tossing standard Newton laws out. It does not generalize to curved space-time, and general Relativity just changes the form of Newton's laws. Momentum isn't just m*v anymore. It's m*v/sqrt(1-v^2/c^2). "Rest" mass curves space, not "Relativistic" mass. In GR, curved space is the expression of gravity. So, no the airplane does not weigh more unless it's accreting a whole lot of bugs or ice.
Quantum physics doesn't change mass. Mass is one of the fundamental parameters along with charge and spin.