While we’re waiting for a current answer, /u/sunagainstgold provided a detailed response to a similar question five years ago

https://www.reddit.com/r/AskHistorians/comments/5zkvxc/where_are_we_on_no_irish_need_apply/

I did not see this post yesterday, so I wanted to address the "dogs" aspect of it specifically as well as the derived mythologies. The "dogs" myth originates in a specific park in the Shanghai leased territory which had some variant on the following wording in its rules from 1894 to 1928: "The Gardens are reserved for the foreign community. [...] No dogs or bicycles are admitted." There is a photograph of the actual sign.

In one English language article written during the time, this sign was shortened to a single, more inflammatory phrase: "no dogs or Chinese". This restatement seemed to be considered apt and was widely adopted in both English and Chinese language memoirs. However, there was never a sign that actually read "no dogs or Chinese", according to a 1995 scholarly article in the China Quarterly.

This phrase then took on a life of its own and became a sort of symbol for the memory of China's humiliation as well as other forms of white supremacy. According to the China Quarterly article, in 1994 two mainland Chinese newspapers ran a denunciation of those who would question the existence of the sign.

Furthermore, the phrase was taken from the context of Shanghai and used even for completely different colonial situations. For instance, former Indonesian president Suharto once asserted in an interview that he saw a "no Javanese or dogs" sign in colonial Jakarta, which is otherwise unattested and surely did not exist. As a low-status farmer, Suharto had benefited from Dutch educational programs set up for the Javanese, but this was a shorthand for describing his distaste for the colonial system at large. (See David Jenkins, Young Soeharto, 2022) I suspect the sort of generic phrase you are thinking of comes from this variant on the myth.