In your example sentence, the actual name of the building is (in English anyway) simply La Scala.
If you use only a proper noun, you do not normally use an article. (Although, stylistically, you might want to in some cases.)
In your example sentence you could say:
The play was presented at La Scala.
You don't need the article because you're referring to the actual name of the building.
If you use just a common noun, you do need to use an article:
The play was presented at the opera house.
The play was presented at an opera house.
But if you want to use both a proper noun and a common noun, then you use the definite article—and only the proper name is in uppercase:
The play was presented at the La Scala opera house.
But let's imagine that the actual name of the building were, literally, La Scala Opera House.
If this were the case, then Opera House (what would normally be a common noun) would actually be part of a proper noun.
Do you put an article in front of it or not?
There is no completely correct rule for this.
In theory, you could say:
The play was presented at La Scala Opera House.
Or you could say:
The play was presented at the La Scala Opera House opera house.
But neither of those necessarily seems correct because of how we're used to constructing similar sentences where this kind of confusion doesn't exist.
I suspect that most people would use an article with only the proper name—because it would seem correct even though, technically, it doesn't fit one of the normal uses:
The play was presented at the La Scala Opera House.
Note, however, that this is not actually the case in your example sentence. The name of the building is simply La Scala. So, refer to the first part of this answer.