Making simple on/off IR servo movement.
Up at the top of your code, you have:

#define START_BUTTON //place code for button // code recieved from start 
button
#define STOP_BUTTON //place code for button // code recieved from stop 
button


A #define needs 2 parts after it. The first is the new name being created, and the 2nd is the definition of the new name.

You should put the value that you expect for the START_BUTTON condition after the name, such as:

#define START_BUTTON 0x48
#define STOP_BUTTON 0x49


Or whatever your IR remote is sending.

Also, you have one too many closing braces (}) at the end.

      [...]
      case STOP_BUTTON:
           Servo1.write(0);
           delay(200);
           Servo2.write(0);
           delay(200);
           break; 
            }  // THIS CLOSES THE SWITCH
     }  // THIS CLOSES THE IF STATEMENT
  }  // THIS CLOSES loop()

  //^servo 2 positions and delays
} // THIS IS NOT CLOSING ANYTHING THAT WAS OPENED