Be sure to get more information from a college's website. In some cases the ACT with Writing can substitute for the SAT Subject tests, and colleges change their admissions criteria all the time.
All the schools below require SAT Subject Tests. Click on the school's name to get a description, admissions data, costs and financial aid information.
Bryn Mawr College
California Institute of Technology (Caltech)
Carnegie Mellon University
Cooper Union (some programs)
George Washington University (required of some dual degree programs only)
Harvey Mudd College
Massachusetts Institute of Technology (MIT)
Notre Dame (Indiana) (home schooled applicants must take 3 SAT II or AP exams)
New York University (NYU) (but ACT or AP exams can substitute for SAT II)
Olin College of Engineering
Rensselaer Polytechnic Institute (RPI) (some programs)
Stevens Institute of Technology (some programs)
University of California -- The state system has changed its policy to read, "While SAT Subject Tests are not required, some campuses recommend that students vying for slots in competitive majors take the tests to demonstrate subject proficiency." You can get more information here.
University of Pennsylvania
Washington and Lee University
The list of colleges and universities requiring the SAT subject tests is constantly changing, so be sure to check with the schools to which you are applying. Also, many colleges will allow an applicant to substitute AP scores or ACT scores for SAT subject test scores.