Note: A phrase's origins are, much of the time, completely unclear. The origins you see listed for many idioms are the plausible theories floating around for how or where they came from, but may not necessarily be accurate. The quotes which have the phrase in them are the oldest written forms of the phrase I could find, but keep in mind that older recordings are probably out there somewhere. If you find an earlier citation than what I have, feel free to let me know!
Also, remember that just because you see a saying in an old book or newspaper, let's say from the year 1893, it does not mean the saying originates from that source. In all likelihood, if an expression is already in use in a book or newspaper, then it's probably older. Nevertheless, these old quotes serve as a way to show the reader how far back in history some of these sayings go.
knowyourphrase.com - Meanings and Origins of Phrases
Being angry about something that happened in the past; holding a grudge.
This common phrase is said to those who remain upset over problems they've experienced in the past.
This phrase goes back to at least the 1800s. A 'chip' can be defined as a piece of timber, or wood. Depending on the amount and size, timber can be quite heavy, and oftentimes people carry heavy things on their shoulders. Well, apparently it became a thing for some people to place a chip on their shoulder in order to show that they were looking for a fight. Yeah, I guess a person, I imagine it was an angry person, would place a 'chip' on their shoulder and then dare others to knock it off. Anyone who wanted to accept the proposed challenge could do so, and then a fight would occur shortly after.
There are newspapers from that century which talk about something along these lines. For example, the Long Island Telegraph newspaper printed on May 20th, 1830, wrote:
"When two churlish boys were determined to fight, a chip would be placed on the shoulder of one,
and the other demanded to knock it off at his peril."
Sharing similar thoughts, the Onondaga Standard of Syracuse, New York, 1830, wrote:
"'He waylay me', said I, 'the mean sneaking fellow - I am only afraid that he will sue me for damages.
Oh! if I only could get him to knock a chip off my shoulder, and so get round the law, I would give him
one of the soundest thrashings he ever had.'"
So perhaps this expression got its figurative meaning of "I'm angry at someone/something" from what's described above.