This switch to me looks like it sends power from the right conductor to either the top or the bottom conductor, depending on whether the door is open or closed.
If you had a working switch and a multi-meter you could easily figure it out.
Probably all it does is when the door is open, the power goes to the light inside the dryer, and when the door is closed, it energizes some relays for the actual dryer machinery. (the motor, heater relays, etc)
Probably you can bypass it by simply putting a spade conductor through probably the wire on the right and to the wire on the left, probably the lower one.
Have the dryer unplugged when you do this. Plug it back in, and the light should not be on with the door open. If you got it the wrong way, the light will be on.
There is a third possibility that you get the wrong input wire (there is no reason the switch has to use a particular terminal on a particular side for a particular function) and you’ll need to try a couple more permutations for it to work.
Obviously you will be able to run the dryer with the door open after you do this.
Don’t get shocked. Also whatever you use to jump the connection, make sure it is a good conductive piece of metal - don’t start a fire. It is possible that all of the power for the dryer motor goes through this switch, given the current rating I suspect it might.
And keep in mind that electrical tape may not be an adequate insulator, it may melt, you might have to use a piece of heat shrink or something over your temporary repair.