Why wouldn’t it? Why would you use the dative here?

I had actually thought you were using the nominative in your first post since ‘tigh’ is used as the nominative and dative in Munster, and so for a moment I assumed that it must be so in Connacht too – though I had always thought ‘teach’ was the nominative there. I got a bit confused. However, I don’t see why you would use the dative and not the nominative when simply stating the name of the pub or shop – e.g. on a pub or shop sign.

Because tigh + name is a petrified phrase.
“tigh” is rather a preposition here, not a noun, used like the French preposition “chez”,
chez Jean = tigh Sheáin.