This may be one of those questions to which we will never find a satisfactory, definitive answer. There seem to be many contenders for the title but none offer any documented proof. My research has turned up the following possibilities (listed in my subjective order of likeliness).
1. "With This Ring" by The Platters (recorded circa October 1967). - Corroboration...
Mirasound [...] on 57th Street [NYC] is the first studio to put in 16-channel recording [...] in the Fall of 1967.
https://www.ilk.org/~ppk/Manuals/Ampex_Case_Histories/Ampex%20Mirasound%20case%20history.pdf
Well, the unit arrived at 10 am in the crate, and we had it connected and checked out in time to do a session at 8pm that same night. The session was "With This Ring" by the Platters, produced by Luther Dixon. So I became the very first engineer to record in 16 tracks.
https://www.historyofrecording.com/ampexag1000.html This second quote is an unverified statement by George Schowerer who was supposedly a recording engineer at Mirasound at the time.
2. "Crimson and Clover" by Tommy James and the Shondelles (late 1968). - Corroboration...
"Crimson and Clover" was recorded in late 1968 [and] is one of the earliest songs recorded on 16-track equipment.
https://en.wikipedia.org/wiki/Crimson_and_Clover#CD_and_single_re-releases This statement is not referenced and therefore somewhat flimsy but is the most authoritative I can find.
3. "Hot Rats" Album by Frank Zappa. (late 1968) - Corroboration...
The album was recorded on what Zappa described as a "homemade sixteen track" recorder; [...] in late 1968.
https://en.wikipedia.org/wiki/Hot_Rats Again, not the best source but the best I can find to quote and there is a reference at the bottom of the article that may support the claim. Which 'song' was recorded first, however, does not appear to have been noted.
Other contenders include the albums "Aoxomoxoa" by Grateful Dead and "Volunteers" by Jefferson Airplane although there does not seem to be any evidence to support their claims. And which of the songs included on these albums was recorded first may never be known.
For an event that occurred over 50 years ago this may be the best evidence that exists; considering that many of the on-site participants have unfortunately passed on.