This is a case where the way English handles the "use/mention distinction" is important. See, without the italics, quotes, or any other distinguishing marks, you could use the words directly just as you normally do and make a grammatical and almost semantically coherent sentence like this (verb bolded):
Other types of residential treatment facility refer to facilities not licensed as a psychiatric hospital, whose primary purpose is ...
The only problem with this would be that the types here do not in fact refer; they actually are.
So what you're really trying to do is say this (verb bolded, elided phrase in brackets):
[The label/codeword/phrase/term] "other types of residential treatment facility" refers to facilities not licensed as a psychiatric hospital, whose primary purpose is ...
Once you realize that there's implicitly a singular noun that introduces the mention of the phrase you italicized, but is elided in the actual sentence, it becomes much clearer how this sentence should be written (verb bolded):
Other types of residential treatment facility refers to facilities not licensed as a psychiatric hospital, whose primary purpose is ...
A phrase that you are mentioning, not using, is considered one phrase. It's singular. So you use singular conjugations with it. (This is a little clearer when using quotes, since that's a more common way to mention a phrase, but italics, bold, underlines, and even different font families have been used for the purpose as well, so there's nothing wrong with the sentence being written this way.)