Moving text on OneNote for iPad
I am a regular user of the current OneNote iPad app.  Your only option for moving the text boxes is to use the iOS method for selecting, cutting and pasting.  There is not currently a way to grab text box handle and move the entire box like you can with the desktop app.  

After you select and cut your text, tap into the area of the page you wish to move the box and then tap and hold and paste.

FWIW, the rumor mill says that Microsoft will release a full-on touch compatible version of the office suite, including OneNote for the iPad very soon.  Hopefully that version will support more of the features of the desktop app like moving the text boxes.

Happy computing.