Picked this from phrases.org.uk.
The Oxford English Dictionary defines the phrase, “get something off one’s chest,” as meaning “to relieve one’s mind by making a statement or confession,” and cites as its earliest example “1902 Daily Chron. 27 Sept. 3/2 The desire is either to deliver a message to the world or to express the individual personality—to ‘get it off your chest’ is the horrid, vulgar phrase.”
Although one need not concur in the characterization as a “horrid, vulgar phrase,” it’s true that it can refer either to something you feel you need to say because it needs saying, or to something you need to say because it has been “weighing on your conscience.”