Continuity mistake: When the press secretary kisses the hand of the alien-dressed-as-a-seductive-woman, the eyeball ring is on her right middle finger but after she knocks him out (or kills him) the ring is on the finger between the middle and the pinkie. In subsequent scenes it's back on the middle finger.
Continuity mistake: When the martians first visit earth in the desert you can see that the shadow from the poles at the red carpet is on the left side, a few seconds later after the silver ramp is unrolled and the martian ambassador walks down it and onto the red carpet you can see that the shadow of the poles is on the right side.
Other mistake: In the scene where the President dies, after the General tells everyone to hush and there are agents/bodyguards slowly approaching the door, watch the door to the room before and while it explodes. There are lines in the door which mark the way in which the door will be blown open.
Other mistake: When those cute little Martians break into The White House and begin to blast away everything, a few of them get shot in the process. One gets shot in the helmet, and collapses to the ground, suffocating. A few minutes later, during the same scene, another gets shot, and likewise collapses. The animation of the two Martians collapsing is the same, just reversed.
Revealing mistake: The majority of the Army rifles and all the machine guns are obvious props, incapable of even firing blanks. If you watch carefully when they're "firing" you can see that the muzzle flash is actually animated and no shell casings are coming out. The sound has been added in post production. This is most obvious when the general is firing a havy machine-gun (with no belt) at the saucer over the capitol.