We want to just move the biztalk message box to different server and let the other BizTalk Database stay in the same box.

What is the best and fastest way to achieve the same with minimum production down time

This article might be helpful.

It's not a difficult procedure, not shouldn't be taken lightly.

You'll get a good understanding of how it works by reviewing the DR documentation, of which as step is essentially moving all the databases, including the MessageBox.

A good place to start: http://msdn.microsoft.com/en-us/library/gg634645.aspx