It is likely an infringement of copyright whether or not you monetize, unless it falls under fair use. Simply responding to the DMCA notice by taking down the video will NOT protect you from a lawsuit (the DMCA limits YouTube's liability, not yours), although it might limit the damages because your video won't be up as long. (But statutory damages are still possible, and they can run up to $150,000 for willful infringement.)
But there's not just civil liability to worry about. Copyright infringement can be a crime. According to 17 U.S. Code § 506 - Criminal offenses:
Any person who willfully infringes a copyright shall be punished as
provided under section 2319 of title 18, if the infringement was
(A) for purposes of commercial advantage or private
(B) by the reproduction or distribution, including by
electronic means, during any 180–day period, of 1 or more copies or
phonorecords of 1 or more copyrighted works, which have a total retail
value of more than $1,000; or
(C) by the distribution of a work being
prepared for commercial distribution, by making it available on a
computer network accessible to members of the public, if such person
knew or should have known that the work was intended for commercial
According to 18 U.S. Code 2319, the maximum punishment for violating the above is between 1 and 10 years, depending on various details.
So, if you consider civil copyright infringement to be "illegal", you are most likely correct. However, if you're only concerned about criminal charges, then your friend is also correct - monetizing could make it criminal.