A furnace is a machine too but they have heat on the Sabbath and holy days, right? They aren't doing the work personally but the machine isn't shut off is it? If that's the case then I don't see why being in an airplane would be that much different.
In the case of a furnace, you have set its desired temperature before the Sabbath and it automatically turns on and off to keep that setting. You don't have to touch it. You are allowed to set electrical things on timers, so you can have lamps come on and off according to timers you set before the Sabbath.
As far as airplane travel, there are many possible violations of the Sabbath involved. You may very well have items you are not allowed to handle on the Sabbath in your luggage, such as electrical items. You can't travel by car to a hotel or to your home on the Sabbath, so you'd be stranded at the airport. You can't use money on the Sabbath, so how would you pay that cab? You can't carry things from one place to another unless special conditions are set up before the Sabbath, so you could not carry your luggage. You could not even carry your house keys! It would be a right mess to arrive after the Sabbath commenced, trust me.