Is it OK to remove a theme designer’s “powered by” link on a WordPress theme you have purchased? 

It depends, some designers and developers specifically request that you leave their footer intact, showing your appreciation and kudos.

Although all themes should be compliant to the GNU gpl license as is required by WP and therefore you are free to manipulate the code to your preference.

You may modify your copy or copies of the Program or any portion of it,
thus forming a work based on the Program, and copy and distribute such
modifications or work under the terms of Section 1 above, provided that
you also meet all of these conditions:

  1. You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.
  2. You must cause any work that you distribute or publish, that in
    whole or in part contains or is derived from the Program or any part
    thereof, to be licensed as a whole at no charge to all third parties
    under the terms of this License.
  3. If the modified program normally reads commands interactively
    when run, you must cause it, when started running for such interactive
    use in the most ordinary way, to print or display an announcement
    including an appropriate copyright notice and a notice that there is no
    warranty (or else, saying that you provide a warranty) and that users
    may redistribute the program under these conditions, and telling the
    user how to view a copy of this License. (Exception: if the Program
    itself is interactive but does not normally print such an announcement,
    your work based on the Program is not required to print an
    announcement.)
    Excerpt from: http://wordpress.org/about/gpl/

Read more:
http://wordpress.org/about/license/
http://wordpress.org/about/gpl/ GNU General Public License