I am working on a formal essay where I have to quote a few American English speakers, but my essay is written in British English. Should I change terms like color to colour?
Usually no. When quoting, it is assumed that you are using the original writer's dialect and spelling, since that is a part of what they wrote. The style guides I consulted agree on that point.
APA has a blog post confirming that spelling standards pertain only to your own manuscript, not quoted material:
The Publication Manual’s spelling guidelines apply only to the original writing in your paper.
For references, keep the spelling in titles and other elements exactly as they appeared in the original. That is, cite what you see.
Likewise, if you quote from the text, keep the original spellings. There’s no need to use [sic], as these are not errors.
MLA suggests that quotes should not be changed unless absolutely necessary, which I would extend to differences in dialect. This is from 1.3.1: Use and Accuracy of Quotations in MLA 8:
The accuracy of quotations is crucial. They must reproduce the original source exactly. Unless indicated in square brackets or parentheses, changes must not be made in the spelling, capitalization, or interior punctuation of the source.
A few pages later, in 1.3.7, an example of keeping punctuation is annotated with "Preserving original spelling see sec. 1.3.1," with an example of keeping a British English spelling:
"How can I describe my emotions at this catastrophe, or how delineate the wretch whom with such infinite pains and care I had endeavoured to form?" wonders Victor Frankenstein in Mary Shelley's Frankenstein (42)." (First italics highlight what was annotated.)
MLA is written for American writers, but the general principle should be true for you: keep the spelling as quoted unless your style guide tells you differently.
You could use [sic] in the quote, this identifies that the mistake is intentional. For example
I don't like that color [sic].
A more detailed explanation can be found at https://data.grammarbook.com/blog/definitions/sic/