Does the rotation of the earth dramatically affect airplane flight time?

During the flight, you need to get up to use the restroom. There's one 10 rows in front of you, and another 10 rows behind you. Does it take longer to walk to the one that's moving away from you at 600 mph than the one that's moving towards you at 600 mph?

No, because you're moving at 600 mph right along with it -- in the ground-based frame of reference. In the frame of reference of the airplane, everything is stationary.

Similarly, the airplane is already moving along with the surface of the Earth before it takes off. The rotation of the Earth has no direct significant effect on flight times in either direction.

That's to a first order approximation. As others have already said, since the Earth's surface is (very nearly) spherical and is rotating rather than moving linearly, Coriolis effects can be significant. But prevailing winds (which themselves are caused by Coriolis and other effects) are more significant that any direct Coriolis effect on the airplane.

When an airplane starts in any direction, its velocity with respect to any reference frame automatically gets the contribution from the moving Earth's surface.

Equivalently, you may look at the whole situation from the Earth surface's viewpoint and then the Earth's rotation is invisible and can't influence the speed and timing of flights, because of the principle of relativity. In this idealized description, there's no difference. This conclusion would be right when we neglected the atmosphere, mostly for e.g. the rockets that spend most of the time outside the atmosphere.

However, there exists the atmosphere and it has winds - whose average speed ultimately depends on the Earth's spinning as well but the dependence is indirect. In moderate zones, the westerlies dominate – winds from the West

http://en.wikipedia.org/wiki/Westerlies

and because the airplane flies in the atmosphere and wants to reach a particular speed relatively to the air mass, it's clear that the speed of westerlies helps you when to speed you up when you fly from the West, and slows you down when you fly to the West. That's why flights from America to Europe (or from Sydney to California) are about 1 hour faster than the opposite flights.

The rotation of earth causes two effects: Centrifugal force and coriolis force.

The effect of centrifugal force is exactly balanced out by the fact that the earth is non-spherical (it bulges at the equator). The whole surface of earth is an isopotential surface with respect to "gravity plus centrifugal force". The downward force that pulls on everything, that people lazily call "gravity", is really "gravity plus centrifugal force" on earth. There is no other special or surprising effects of centrifugal force on earth. Things get pulled down towards the ground like we intuitively expect. [Update--As pointed out in the comments, gravity is weaker at the equator, but only by a fraction of a percent; I doubt that measurably affects airplane speed.]

Coriolis force has a very important and very indirect effect on air travel because it alters winds, weather, and in particular the direction of the jet stream. (See Luboš's answer.) As for direct effects on the airplane, it is negligibly small. The plane experiences a force pushing it rightward (in the northern hemisphere), about 300X weaker than gravity or more (if I calculated correctly). The pilot steers very slightly leftward to compensate, maybe not even consciously.