Is it permitted to play an acoustic Gramophone on Shabbat? An acoustic gramophone is a non electric device that plays disc records by having them rotate under a point.
It is forbidden to sound musical tones on Shabbat using a musical instrument. The issue is not about using electricity but so one does not come to fix the instrument since repairing an instrument on Shabbat is a Torah transgression -- either boneh (building) or make be patish (making an item usable).
The Rambam writes (Hilchot Shabbat 23:4)
Accordingly, it is forbidden to sound musical tones on the Sabbath, whether using a musical instrument - e.g., a harp or a lyre - or using another object.
This is codified by the Shulchan Aruch in OC 339:3 in the context of clapping hands
We do not clap nor to we slap our hand to our thighs nor do we dance. This is a rabbinic decree lest one come to fix a musical instrument. [...] All of this and anything similar is forbidden, based on a rabbinic decree preventing the fixing of a musical instrument.
Since it is forbidden to use instruments, even touching them is forbidden because of mukze - either kli shemelachto le-isur (a utensil used for forbidden actions) or if very valuable machmat chisaron kis (fragile utensils).
How does the disc rotate? Probably by turning a crank, which seems no different from any number of work activities such as operating a spinning wheel or riding a bicycle. No cranks are mentioned in the Bible, so the prohibition would necessarily be rabbincal. I do not know if any rabbinical authority has specifically addressed the issue of gramaphones, but operating them by hand seem logically in involve "work."
See this article for some interesting rabbinical debates about the permissibility of music-listening on Shabbat.