The Firefighter career is known to be very buggy, however here are a few things to try:
1. Switch to the family that's having the emergency via Edit Town and then switch back to your family. This is a way of resetting them.
2. Reset Everything via Nraas
Reset Everything -
Selected Sim > Nraas > City Hall > MasterController > Town > Reset Everything
3. If need be, you might try quitting the job and then rejoin the career. You will loose your job level, but that you can cheat back up with Nraas MasterController if desired.
Change the Career Skill Level -
Click on the sim, then NRaas > Master Controller > Intermediate > Career > Level > (Choose
the level you want).