What is the origin of the phrase "to go apeshit"?

Note that "-shit" is often used as a slangy intensifier: bullshit, chickenshit, batshit. The enduring popularity of bullshit (which in the sense of "falsehood"/"nonsense" is actually of much more recent vintage than the equivalent bull, despite what one may assume) probably inspired the others.

Apeshit can therefore be traced to ape (in the "going ape" sense), just as batshit can be traced to batty or bats. All four basically mean the same thing.

The OED lists it as just a variant of "to go ape", with the earliest citation of both being a single entry in a 1955 edition of "American speech", where they are both listed as "air force slang". The OED also says "See American Speech (1961) XXXVI. 150 for an account of the phrase's development." - that seems to be available only behind a pay-wall.

I think it may be more specific than just some aggrieved simian. Apes in captivity famously express their displeasure (with each other, with zoo personnel, and with zoo visitors) by flinging feces.

The more polite way to express the same idea is "go bananas". As the late great George Carlin pointed out, bananas are apeshit.