Can I use unlicensed music in a birthday video and take money for my editing work?
I am not a lawyer and it may depend on where you are located, but generally speaking, even if you weren't being paid, this is technically a violation of copyright that you could be sued over.  It is highly unlikely that anyone would come after you for it if they only use it privately, but if it was, say, published to youtube, then there would be more chance of issues coming up and you would still be the person who sold them the video.  You would probably be liable under most jurisdictions copyright laws as this is a pretty major situation that copyright is specifically designed to prevent.  

The intent or profit is generally irrelevant to most copyright law.  All that generally maters is the amount that it should have cost to make use of it (the actual damage is the money the author didn't get that they would have gotten for the usage).  So even if you gave it away for free (like many file sharers do), the damages are based on what the licensing for that should have been, plus a punitive penalty.  Whether you give away the work or sell it doesn't matter because it is still a work that the creator should have been paid for as it is their work.