"move rapidly," 1852, of echoic origin. Meaning "close with a zipper" is from 1932. Related: Zipped; zipping.
"to close or fasten by means of a zipper," 1932, back-formation from zipper (n.). Related: Zipped; zipping; zipless.
1963, in U.S. postal ZIP code, an acronym for Zone Improvement Plan, no doubt chosen with conscious echo of zip (v.1). Alternative post code is attested by 1967 in Australia.
"sound of something moving rapidly," 1875, imitative. Zip gun "homemade pistol" is attested by 1950.
"zero," 1900, student slang for a grade of zero on a test, etc.; of unknown origin; compare zilch.