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.
Other mistake: The first time Rachel brings the team coffee, she has six coffees on a tray. There's five cappuccinos and one regular coffee for Harry. On the tray there is five coffees on one side and one coffee at the other side, obviously being Harry's regular coffee, so that Rachel would be able to tell the them apart. But when she gives Harry a coffee, she gives him one of the five standing together, so he obviously got one of the cappuccinos.