Nope, turning off the stove just means they hadn't shut it off after cooking a meal so you'll see a pot or pan on top of the stove cooking away.

Leaving it on won't start a fire just like all the other actions.. a fire is a completely random event just like the other appliances breaking down. If you do catch a fire, there's much more to it than just putting your person down on the stove, they'll just point out that the oven's on fire.