I have two non-iron button down type shirts, and my husband has several...the no-iron aspect doesn't seem to work without putting them in the dryer.
They come out of the dryer in near-pristine condition - requring no ironing at all (unless you wanted a crease down your sleeves, or perhaps just a touch near the collar), and they don't seem to shrink either. One trick is to make sure you retrieve them out of the dryer immediately...or even before the dryer is finished, and hang up right away. My dryer has a setting called "wrinkle prevent" which means that if I forget to retrieve items right away, it'll toss them around every few minutes (air fluff only) to keep them from wrinkling. Since my washer and dryer are down in the basement, I'll often set a kitchen timer to go off a few minutes before the dryer finishes, and I'm down there, hangers in hand before it stops.
I don't put any other shirts in the dryer though - just the non-iron ones.
Just a note: my Mom bought a couple of non-iron shirts too...and she couldn't figure out the 'wrinkle free' aspect either, until I explained that you had to be there to get them out of the dryer right away, and that you don't dry them for very long. Maybe 20 minutes at the most?
If you're nervous about trying it, you might consider buying a less expensive no-iron shirt, and seeing how it does in the dryer. My Mom got the men's Kirkland no-iron shirts for my stepdad, after I raved about them. The two I have are from Eddie Bauer (white and a french blue colour). One last thing: I wash these in cold water ...not sure if that makes a difference, but perhaps it does? Oh, and for whites I throw in Oxy powder..even though I'm just washing them in cold water. Oxy powder works beautifully on whites.