Help with process interrupters

I'll preface this by saying I'm completely new to Arduino and basically have no idea what I'm doing. I can build simple circuits, but I haven't programmed anything except for some HTML several years ago.

I'm working on a project where I set a timer, and before the timer runs out someone needs to either open a switch/cut a wire/remove something from the circuit in order to stop the timer. Remove the wrong one, and the timer speeds up. Are process interrupters the proper way to accomplish this, or is there a better way? And if they are the best way, how would I wire/program multiple versions into one system? So far the only things I'm coming up with in my head simply cause a break in power to the LCD/and or light/alarms until the component is replaced, but that's not what I want to accomplish.

Are process interrupters the proper way to accomplish this

No.

And if they are the best way, how would I wire/program multiple versions into one system?

It is straight forward to do one system but for multiple systems you need to use a state machine.

See my
http://www.thebox.myzen.co.uk/Tutorial/State_Machine.html
Or Robin2's several things at once
http://forum.arduino.cc/index.php?topic=223286.0

Just poll the extra wires. Use the internal pullups, and wire the pins to Gnd with the to-be-cut wires.
Then they normally read low - if they read high then someone has either opened a switch/cut a wire/removed something from the circuit, do something in code.

Google "airsoft bomb"