Can you clarify what you mean by debug message bar? Do you mean the status display below a node? If so is it a function node? If so then write an empty string to it.
That would be a handy feature - could save quite a few clicks on the clear debug button
Moved to feature requests
Short answer is no, but - the sidebar will scroll so you can always see the latest messages. If you want to separate "runs" more clearly you could write a node.error message at start time to colour it red in the side bar.
There is a keyboard shortcut for clearing it if you don't want to do the mouse miles.
Perhaps a 'Clear on Deploy' option would be useful, it's what I usually do anyway, click the Debug dustbin on the way up to the Deploy button.