I researched this carefully while planning out my mod path.  Some get no gains with a DP while others get small gains.  But they are very SMALL gains....only a few HP.  Cobb, for example, got +7 HP on their OTS tune for "stage 3" after adding a DP and exhaust.  I figure 2 of that is the DP, 2 the exhaust, and maybe 3 the tune tweak itself.  The thing is, on our cars, even if it does help it only helps a little.  I decided against these "stage 3" mods because 7 HP was not worth $1000 to me.  Plus I didn't want the noise.  Plus I install myself which isn't exactly fun, or pay someone more money to install.  IMHO that money can be used for other things including a custom tune which will gain me at least 7 HP or maybe more.  I recommend "stage 2" mods and then a tune and some bracing.
As for using stage 3 tune without stage 3 mods, don't try to run your car on a map for parts you don't have - don't cheat the system, it isn't good for your car.