The Best time to take SAT Subject tests
Are you taking Precalculus or AP subjects ? Or have you scored 1400 above on SAT ?
Then you might need to prepare for SAT Subject tests now. The best time to take SAT Subject test is usually at the end of the school. You can register for the one in May or June at SAT.ORG.
SAT Subject Tests Requirements by School
Non-exhaustive list of top schools' SAT Subject Test requirements
Amherst: Considered
Barnard: Considered
Brown University: Recommended, 2 tests
California Institute of Technology: Required, 2 tests (Math Level 2 and one science)
Carnegie Mellon University: Recommended, 2 tests (except for the College of Fine Arts)
Columbia: Considered
Cornell: Required, Arts & Sciences and 2 tests for Engineering (Math Level 2 and one science)
Dartmouth: Recommended, 2 tests
Duke: Recommended, 2 tests
Emory: Recommended
Georgetown: [Strongly] Recommended, 3 tests
Harvard: Recommended, 2 tests
Johns Hopkins: Considered
MIT: Required, 2 tests (math and one science)
Northwestern: Considered
NYU: Not required
Pomona College: Considered
Princeton: Recommended, 2 tests (Engineering: math and chemistry or physics)
Rice: Required, 2 tests (related to intended field of study)
Stanford: Recommended, 2 tests
Swarthmore: Considered for Non-Engineering; Recommended for Engineering (Math Level 2)
Tufts: Not required
UC Berkeley: Recommended
UCLA: Recommended for Engineering, 2 tests (Math Level 2 and science)
UChicago: Not required
UNC Chapel Hill: Considered
UPenn: Recommended, 2 tests
WUSTL: Considered
Wellesley: Considered
Yale: Recommended
SAT Subject Test Reality Check: How much do they matter?

Many elite universities have shifted from "requiring" to "recommending" the tests in recent years. However, for most students, particularly those who do not come from disadvantaged backgrounds, "recommended" means you definitely should submit strong Subject Test scores. However, they do count for less than SAT/ACT scores.

How do strong SAT Subject Test scores help your application?
If you claim to have a strong interest in a particular field, admissions officers use subject tests to see if you are actually interested in it or if you're merely saying it.
It is another way to show your intellect. Moreover, because a 600 vs an 800 is a much bigger gap than a 4 vs a 5 on AP Exams, performing well on SAT Subject Tests is often a better sign of academic performance than even AP exams.
In the words of admissions officers themselves (from Inside Higher Ed):
"James Colman, senior associate director of undergraduate admissions at Georgetown University, said that 'the scores still have real value [and] that committees find scores helpful in weighing applications...this may be especially important for students who study languages but don't participate in Advanced Placement programs, or for homeschooled students.' More broadly, he said 'Subject Test scores are helpful in light of grade inflation.'"
"Jarrid Whitney, executive director of admissions and financial aid at Caltech, which continues to require subject tests, said that the results provide 'a more granular perspective' on applicants' math and science abilities, which are 'critical,' given Caltech's emphasis and rigor."
Typical SAT Subject Test Scores for Top Schools
Although selective schools tend not to publicize the average SAT Subject Test scores of their accepted applicants, you should aim for scores in the 700s, particularly 750+. This translates to 80th percentile and above, and 90th percentile and above.

In the past, MIT has stated that their admitted students score between 730 and 800 on math Subject Tests, and 720 and 800 on science Subject Tests. This means that an 800 on math and science for MIT and other top engineering schools is typical, and should be your goal. Princeton admitted students generally score between 710-800 on Subject Tests. At Harvard, a 700 and better on a language Subject Test allows you to pass their language requirement.