Cannot move email items from inbox to other folder

Try resetting your Gmail Account. First close your Mail App completely from the app switcher window by double clicking the Home button and swiping up the Mail preview pane up until it disappears from the screen. You may have to look for the Mail pane by scrawling to the left. Then go to Settings > Mail, Contacts, Calendars > locate Gmail and click on the account. On the next page Delete Account. Then do a forced restart. Hold down the Home and Wake/Sleep buttons at the same time for about 15-20 seconds until the Apple logo appears. Then go back to Settings > Mail… and add the account again.

How are you trying to do this and which provider do you have? With a Yahoo iMAP account I can select the email, the folders, and then drag it to the proper folder. What are you trying on yours?