This is covered quite thoroughly in a column in The Globe and Mail by Warren Clements. Summarizing:
- The romantic sense of crush was first recorded in the 1884 journal of Isabella Maud Rittenhouse.
- According to Eric Partidge, crush might have been a variation on mash, since by 1870 mashed was a popular way of saying flirtatious or head over heels in love, and to crush something was to mash it.
- How did mash acquire that meaning? The Oxford English Dictionary shoots down one line of speculation found in an 1890 slang dictionary: the suggestion of a Romani word mash or masherava, meaning to allure or entice.
- Ernest Weekley’s 1921 Etymological dictionary of modern English observed that mash was “regarded as spoon-diet,” the sort of mashed pap one would feed to a person who couldn’t chew or tolerate regular food. He drew a link to the word spoony.
- To be spoony on someone had, since the 1820s, meant to be foolishly or sentimentally amorous, drawing on the earlier meaning of spoony as simple or goofy. To spoon with a lady was to make goo-goo eyes and whisper baby talk to her in the hope of future considerations.
This origin tale requires a couple of leaps of faith – crush to mash, mash to spoony – but as a story it hangs together.
The 1884 date is confirmed via etymonline:
1590s, "act of crushing," from crush (v.). Meaning "thick crowd" is from 1806. Sense of "person one is infatuated with" is first recorded 1884; to have a crush on is from 1913.
A relevant entry from the Routledge Dictionary of Historical Slang by Eric Partridge read:
mash. A sweetheart: 1882. Also MASHER. 2. A dandy: from ca. 1883;
v.t., occ. v.i. To court or ogle or (attempt to) fascinate a girl or a woman; not often used of a woman 'bewitching' a man: 1882, Leland, 'These black-eyed beauties' — Gypsies — 'by mashing men for many generations ....'; ob. Prob. ex the S.E. sense, to crush, pound, smash utterly, but perhaps, as Leland suggests, ex Romany mash (masher-ava), to allure, entice. Orig. (ca. 1860), U.S.
There are a number of other variants and phrases based on mash with similar connotations.
ODO's entry for masher corroborates this sense:
a dandy of late Victorian or Edwardian times.
North American informal a man who makes unwelcome sexual advances to women.
late 19th century: probably a derivative of slang mash 'attract sexually', 'infatuation', perhaps from Romany masherava 'allure'
(Unless the OED differs drastically from the ODO in this matter, the origin stated above contradicts Clements' claim that the OED does not subscribe to this theory.)