When Ted Stryker is attempting to land the shuttle, he makes reference to the events of Airplane! and talks about landing a Boeing 767 in Chicago. In "Airplane!" he was landing a Boeing 707. The Boeing 767 was not in service until 1982.
When Ted lands the shuttle on the moon, it bypassed the moon colony and crashed as it did it spun around so the nose was now facing to the Base, later in a long shot, when everyone was jumping out of the shuttle the long camera shot shows a long deep trail left by the shuttle crash, yet there is no spin out trail and the shuttle nose is facing away from the base
Right after they blow the computer, Ted tells Elaine she'd better check the readout. When she gets up, the back of her shirt is untucked. A moment later, when she is standing at the engineer's station, the back of her shirt is tucked in.
Just before take off, the Navigator and the First Officer were introduced as Unger and Dunn respectively. However, during the computer malfunction scene, their names seem to be switched and the Navigator was being called Dunn and the First Officer was being called Unger.
During its runaway course, the passenger shuttle travels through an asteroid belt just before heading straight for the sun, implying that there is a major asteroid field between Earth and the Sun. There are no asteroid fields of any kind between Earth and the Sun. The main asteroid belt is located between the orbits of Mars and Jupiter. There are also several other minor asteroid fields, but all of them are located near the edge of our solar system and/or beyond the orbit of Neptune.
During the shots of the shuttle in space, at various times a black bar seems to block out the stars as the shuttle flies by them (happens to the left, right, and under the shuttle). Obviously an arm holding the shuttle.