As you know var_dump() in addition to value show its data type and length.
Is there any way to log its output to
You are over complicating what is a simple issue. Firebug (and any other console/dom log viewer is for viewing client side output. PHP being server side and doesn't make much sense pushing to the console log.
With that said, if you REALLY wanted to pipe a server-side output to the console log you should convert that output into json and pass it onto the console log. If you simply want to output variable values on a life site without people knowing that you are working on it (and you shouldn't be working on a live version anyway but that's beside the point) why not pipe the output to a file and read that output however you like, you could even use ajax to pass the dump off to the log via jquery.
The point I am trying to make is...you are over-complicating what you are trying to do.