Q. “. . . go to high school in Washington[, D.C.].” Is the final period necessary? Delete it? (Yes, it’s the last of a longer quotation.)
A. In your version, where you’re using brackets to supply not just the abbreviation but the comma that would normally go with it, you don’t need that final period; we can assume that your bracketed interpolation includes all sentence punctuation, including any final period. And that’s what we might expect if you were supplying the end of a sentence that’s missing or illegible in the source. In other words, your brackets restore the end of a sentence that would normally be punctuated like this:
“. . . go to high school in Washington, D.C.”
But if you’re simply clarifying for readers that the text is referring to the district rather than the state, don’t add that comma. Instead, put “D.C.” in brackets and add the sentence-ending period:
“. . . go to high school in Washington [D.C.].”
That extra period is needed for the same reason you’d add a period to the end of a sentence like this one (from CMOS 6.13):
His chilly demeanor gave him an affinity for the noble gases (helium, neon, etc.).
But there would be no periods in an initialism like DC in current Chicago style, so you’d normally write this:
“. . . go to high school in Washington [DC].”
See also CMOS 6.110 (which has a similar set of examples but without periods) and 12.70–74 (on editorial interpolations and clarifications).