Define "on duty."
At UA, I think (depending on what you mean by "on duty") the answer is no. They don't care where you are when you get your assignment, as long as you are on time to check-in. There is generally a four hour call-out period (i.e., if you get a call for a trip less than four hours before departure, you will not get in trouble if you use your best efforts to make the trip, at least in theory).
I heard that USAirways IS the only airline that makes their reserves stay in base while on duty and I am just trying to confirm that.