In a recent chess game, commentator Peter Svidler said: " Beatings will continue until morale improves". The other commentators seem to know - and enjoy - the quotation. This is a reference to what?

The saying is generally used sarcastically or comically. It refers to authoritarian leadership when demands are made of people be happy, joyful, accepting, or grateful, etc. when there is no reason to be so and usually the opposite is true, the people are miserable. So in the usual authoritarian manner the leadership beats people to achieve the desired result, thus "The beatings will continue until morale improves".

https://quoteinvestigator.com/2020/07/15/morale/

If my memory is correct, there was an old I Love Lucy episode (Lucy and Ethyl in the chocolate factory) which used a version of this saying indicating it was fairly widespread in the 1950s.