Pressing the DB Web Admin button in Plesk to load phpMyAdmin results in nothing happening.
This occurs because the Web Admin panel for databases (phpMyAdmin) loads in a pop-up and most modern browsers now block pop-ups by default. Thus, your browser is blocking your access to the admin panel. To fix this, allow pop-ups in your browser for the page. Here is a list of browsers with links to their help docs describing how to allow pop-ups:
Google Chrome - http://support.google.com/chrome/bin/answer.py?hl=en&answer=95472
Mozilla Firefox - http://support.mozilla.com/en-US/kb/Pop-up%20blocker
Internet Explorer - http://windows.microsoft.com/en-CA/windows-vista/Internet-Explorer-Pop-up-Blocker-frequently-asked-questions
Safari - http://support.apple.com/kb/ht2521 (Scroll to the section on blocking pop-up windows)