PrestaShop No Save Product
Probably you may have a JS error preventing the script to load correctly.
Also a bad module configuration may be preventing the tab loading and if all tabs are not loaded and the buttons won’t stop spinning.
I’d suggest to disable the non native modules in the Advanced Parameters > Performance and check the product edit page again.
If it works then you may disable the option and disable the latest modules you may have installed.
Then activate them one by one and check the edit page every time to find the culprit.