I am currently a sophomore in high school taking Latin III. I plan to drop language after this year and take another science course instead.
How much of a negative impact might this have if I were to apply to schools like MIT, Berkley, UMich Engineering, etc.?
Look at their common data sets and see how many years of foreign language they require/recommend. For example, MIT recommends at least 2 years of foreign language, so it should be okay if you drop Latin III as long as you are replacing it with other meaningful classes (which it sounds like you are.). http://web.mit.edu/ir/cds/2015/c.html