Beyond a Tribble, Andorian etc there wasn't much in the way of visual call backs to the TV show (phasers/props, uniforms, nacelles, sets etc) like there was in later Trek (TNG/Relics, DS9/Tribbles, FC/Phoenix, ENT/season 4, and obviously the JJ films. plus Beyond did it with ENT). Obviously the ships/sets were all similar/updated but there wasn't much in actual TOS style ships/props etc. Not thought too much about where stuff could've been seen but TMP was set not that long after the 5 year mission so you'd expect the odd uniform, prop, ship popping up maybe (the Directors Cut CGed in a TOS style shuttle taking off at San Francisco). Maybe a few prop things in the Botany Bay, a TOS ship in spacedock in III (think either the Phase 2 or Planet of the Titans Enterprise can be seen almost hidden) , and VI could've done with a few more visual nods esp as it was the 25th (maybe he alien Kirk fights could've been a Gorn! -obviously updated abit)
It seems as though the movies were deliberately trying to get away from the visual look of the series as if to say hey forget the cheap looking TOS this is what Trek is supposed to look like (*obviously the TNG films didn't have that issue). Sure I read somewhere that at the time of the first few movies it was considered by some as that's what TOS always looked like -ships, Klingons etc but the budget wouldn't allow for it and it wasn't until later (Relics?) that TOS 'look' was fully acknowledged which eventually led to even bother with the Klingon explanation.
It seems as though the movies were deliberately trying to get away from the visual look of the series as if to say hey forget the cheap looking TOS this is what Trek is supposed to look like (*obviously the TNG films didn't have that issue). Sure I read somewhere that at the time of the first few movies it was considered by some as that's what TOS always looked like -ships, Klingons etc but the budget wouldn't allow for it and it wasn't until later (Relics?) that TOS 'look' was fully acknowledged which eventually led to even bother with the Klingon explanation.