How do I set a continuous servo to turn on until a switch is flicked?
Your are not debouncing your input buttons, which means that digitalRead(Closed) (and Open) will be HIGH in one execution of loop and in the next pass, 3 millis later, it will be LOW.

The servo probably need over 50ms to complete the action (you don't said what servo are you using).

You need to wait for the action to be completed before trying to move it in another direction.

You didn't post your schematic, so may be you wired things wrong. How can we know? 

Usually the first step when you find a problem is add Serial.print to know what is working and what not, and then correct the problem.