I recently discovered a great trick to get a snap off. Take a pair of regular pliers and apply them to the cap, being careful not to pinch any fabric in them, then squeeze the cap as hard as you can. This will cause the post of the cap to break off from the cap. You can then pull the other side out. You will have a small hole from the post but better than a ruined diaper.
If you have any questions I could take some pics to show what I am talking about.
If it is a resin snap, you can either put it in your press sideways and crush it, or my preferred method is to take DH's drill and drill a hole in the middle of the stud. It pops right off. There is usually a little hole, but the replacement goes right in it.
I do this so often that I keep a Dremel in the drawer below my snap press! The little drill bit is perfect for seperating the pieces without destroying the fabric. Plus you can reuse the socket or stud that way, so you only waste the cap.
Ihave tried popping it off by putting it cap side up in the snap press and it never works for me! I've tried smashing it sideways, I've tried prying it apart...I ruined a couple diapers trying to do that! I wish it would work like that for me, it seems so much easier!
I use the snap press to get it off, but it's a 1/2 ton arbor press, so lots of power. Unfortunately, usually when I have a snap issue, it's because I pressed too hard, and there's now this BIG hole where the snap is, rather than a little one. :P