You need to fold over a corner or edge just a little bit and then make a small tare in the paper part. If you tare carefully, you should be able to only rip the paper and not the iron-on part.
Then grab the edge of the paper and slowly pull across the design while keeping the design against the table (better leverage).
Also, I do not pull the paper off of the very edge of the iron-on.
After you have carefully trimmed out your design, look along the edge of the design for any part that has a slight separation and gently pull the paper off starting from there.
Even if you cannot find a slight separation, if you rub your fingernail along the edge of the paper/iron-on, you can usually get the pieces to separate just enough to get your nail and then finger in between the pieces.
Make sense?