What can happen to your car if you run over a traffic cone?

It would be extremely unlikely for any permanent damage to be done. Traffic cones are made on purpose not to hurt cars if they get run over.

Not really true - I was travelling about 60 mph late at night one day and discovered a traffic cone lying sideways in the middle of the highway. For some reason, I thought I had enough undertow to run over it (not!) - next thing I know - there's this loud scary "wham" and from then on I have this funny buzzing sound in my car that turns off and on. Worse, the cone broke my engine mount and caused a rather noticable dent in the underpart of my Honda Civic. Costly - am checking out how much to repair. I think that it shifted my hood as well as it is not aligned correctly anymore....