NO. When you upgrade to a Professional account, it is your YouTube channel that is upgraded. You can install Tube Toolbox on any computer and log into your account from that point on with no limitations.
Here are some Solutions for using Tube Toolbox on multiple computers:
How To: Move Tube Toolbox to a new computer or transfer data to another PC
How To: Transfer Data between Computers
How To: Run Tube Toolbox on multiple computers