The copyright owner is entitled to recover the actual damages suffered by him or her as a result of the infringement, and any profits of the infringer that are attributable to the infringement and are not taken into account in computing the actual damages. In establishing the infringer’s profits, the copyright owner is required to present proof only of the infringer’s gross revenue, and the infringer is required to prove his or her deductible expenses and the elements of profit attributable to factors other than the copyrighted work.
Second, some forms of fair use – which permit use of copyrighted material – do take into account whether someone was aiming to make money from it (though it’s not a “all noncommercial use is fair game” sort of thing):
Notwithstanding the provisions of sections 17 U.S.C. § 106 and 17 U.S.C. § 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include:
the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
Ehhh…sort of.
You’re right to the extent that it’s not a straight “copyright infringement requires that the infringer profit”, but in US copyright law:
First, the copyright holder can take profits that are made by the infringer:
https://www.law.cornell.edu/uscode/text/17/504
Second, some forms of fair use – which permit use of copyrighted material – do take into account whether someone was aiming to make money from it (though it’s not a “all noncommercial use is fair game” sort of thing):
https://en.wikipedia.org/wiki/Fair_use