Here is a recipe similar to the one I use:
1 cup milk
4 tbsp flour
1 cup sugar
1/2 cup shortening
1/2 cup butter (or you can use all butter or all shortening)
Do you think this needs refrigerated?
If it's cooked, I see no reason why it should be.
My boss had to make this frosting a few weeks ago for a wedding, and if she hadn't refrigerated it, it would have been a sloppy mess of a cake.
So does it require refrigeration? Possibly not, you never know by that amount of milk. Is it a good idea to refrigerate? I think so. It obviously can't hurt it.