How do I move a contact on the iPhone from one group to another?
I had the same problem. I found this page looking or a solution. I wasn't satisfied with the above. So I did the following:


Email the contacts to yourself (share contact).
In the contacts app, make sure you are in the proper group (for me, my gmail account with exchange)
Import the contact. It should be adde to the group you selected in the previous step. 
Now go check the contact is in the correct group, and if so, delete the one in the old group.