When Charlie asks how many insults about his own nose he's delivered in the bar someone yells, "14, Chief!" when in actuality it was 18 or 19. (One was deleted in the TV version, resulting in the different totals.)
When Roxanne is hiding outside the firehouse, and C.D. offers her a blanket, she says "No - I'm fine standing naked out here in the cold", she later says that she was being ironic. Except she wasn't, she was being sarcastic.
There is an establishing shot of a cat up a tree with the camera high up in the tree looking down. The cat is obviously fake and on the wide screen version we see a crew member's hand come in from the lower left to steady the fake cat. We know it's not the fireman's hand because he hasn't gone up the ladder yet.
In explaining quarks to C.D., Roxanne (an Astrophysics PhD student who should know better) mentions that there are six types of quarks and that the "top" and "bottom" are the most common kinds. She meant "up" and "down" (which form protons and neutrons). Experimental evidence for the predicted "top" quark was not announced until 1995, 8 years after the film's writing.
After rescuing the cat from the tree, the firefighters take the truck back to the firehouse with the ladder still up. C.D. runs after the truck and sees that they ran the ladder through the wall. The ladder is seen coming off the truck at a very low angle and straight back, but is sticking out the side of the firehouse at a very high angle.