What are the chance a curling iron will start a fire if left on all day?
Do the chances increase if it is by clothes? How long will it take to start a fire, if it will?
Lots of variables. If your electrical system is in good shape, and the curling iron is not near any flammable objects, and in good shape, then there shouldn’t be a fire at all.
But there are a lot of things in the chain that could go wrong.
I would suggest you go home and turn off your curling iron 
They have a thermal fuse in it that will shut it down before it gets hot enough to start a fire. And if it were to start a fire it would have an easier time if it was close to flamable objects. Why tempt fate?
Awhile back, a local nekkid dancer reportedly sat on her curling iron. For a week or so she was showing her fans a “+” sign. So you see, curling irons are not without hazard.
I remember house fire in my town a few years ago that was started by a curling iron left on, on a bed. It was dropped there as the girl got ready for school in the morning, and the fire was discovered in the early afternoon.
I’ve been to 3 building fires in the past 5 or 6 years that were caused by curling irons, all of them having been left either on a carpeted floor or a bed whilest the last user left for the day. It does happen…