There's a lot of stuff I'll pay a bike shop to do. Repacking bearings, replacing cables, truing a wheel (it's an art - an art I'm not willing to constantly practice at enough to get good at), replacing brake pads (tried once, couldn't seem to get the angle right, maybe I'll try it myself again some day), etc.
I'm kinda lazy, I don't really like working on my bike. I usually even pay them to clean the chain (all that hassle to get rid of the used oil properly...).
But I change my own flats because the only way it would be easier and faster for a shop to do it is if my tire got a flat as I was putting my bike in the bike rack at the bike shop - on a day they weren't booked. I can change a flat in the amount of time it would take me to put my bike in the car and drive the bike to the bike store. And that's not counting the time it takes to unload it, talk to the mechanic, wait for him to get to it, wait for him to do it (and that's if they're not already booked or something), pay, load it back into my car, and drive home - it's MORE time consuming and difficult to get the shop to do it than it is to do myself.
It's just faster and easier to do it myself, especially when you consider that I really need to have the supplies (spare tube, pump, patch kit, etc) on my bike for fixing a flat if I got one in the middle of nowhere, and it's a no brainer.
Heck, the next time I get new tires I'm tempted to pay them to put them on (what a pain in the ... last time, new tires are much harder to get on and off than tires you've used a bit). But fixing my own flats just seems like the easier, lazier option to me.