Continuity mistake: When they are driving up the mountain to get to her kids, they cross a river and they get hit on the passenger side of the truck shattering the window, but later when they show them going up the mountain there is a new window there.
Revealing mistake: After Dante's Peak explodes for the final time, the pyroclastic cloud rolls down the mountain toward the town. Harry shouts "Don't look back, kids, don't look back." and then we are shown a wide shot of the truck zooming down the ash-choked streets. Unfortunately, it's painfully obvious that this is either a miniature or CGI shot, because the truck isn't leaving any tracks behind it or trailing any clouds of ash despite its high speed.
Factual error: In the scene where Pierce Brosnan hot wires the pickup truck, it is clear the truck has a locking steering wheel and gearshift. Even if he connected the correct wires to start the truck, he wouldn't be able to move or steer it.
Trivia: When we see Paul in the van as it is taken over by the overflowing river, just as he hits the water, we hear the Wilhelm scream.