The short answer is it depends on what you're ordering, but in most cases yes, you should place the order about 15 minutes before you arrive if you don't want to wait the extra time. Here's what has worked for us:
In the mobile order screen it asks you to tap the button that says, "I'm Here, Purchase My Order." If you are ordering something that isn't inherently gluten feee and needs to be cooked separately, like chicken strips or pizza, I would suggest tapping that button as soon as you order and then making your way to the QS location. If you are ordering something that just needs to be composed, such as a bowl at Satu'li, tap the button when you actually are at the QS location, since the items in the bowl are already prepared.
We have found the best QS with gluten free options for mobile ordering in each park to be:
MK--Columbia Harbor House, Cosmic Ray's, Pecos Bill, Pinocchio
HS--Backlot Express, Rosie's, Pizza Rizzo
AK--Flame Tree, Pizzafari, Restaurantosaurus, Satu'li Canteen
EPCOT--Electric Umbrella (however this QS is closing soon, so not sure where to mobile order when that happens)