I've found that I need to simply get to an available text entry box, hold my finger down in the area until the option to "Paste from clipboard" appears which brings up the current list of items stored in the clipboard (note that, for me on my Droid X, the list has a lot more stored than simply ONE item to be overwritten so the suggestion to "just copy something else to replace it" will not work for me).
At that point, I press the Menu button and that brings up the single option to "Clear the clipboard" which I select and everything is deleted.
The clipboard information MUST be stored in a text file somewhere. This is indeed a security issue and there must be a more secure way of dealing with this situation especially if the clipboard information is stored in clear text.
Only way to do this is to paste the password where it needs to go and to immediately go through the motion to paste again but instead select the option to clear the clipboard.
If you use KeePass, it will clear the copied user name and password after a set time (I set mine to 1 minute which is enough time to copy/paste the pertinent information.)
I hope some of this helps,
Kevin