How is verbal irony used in "who's on first base?" How does that add to the skit and how is that great humor?
In order to appreciate the joke, you need to know the facts. Here are the basics.
"Who" is on first base.
"What" is on second base.
"I don't know" is on third base.
"Why" is in left field.
"Because" is in center field.
"Tomorrow" is pitching.
If you know these basic facts, you pretty much get the joke, because there is complete confusion. The names of the players in the game are common interrogative adverbs and conjunctions. So, when you are talking to someone and you use these words, the automatic assumption is that these words do not refer to names of people, but parts of speech that are common. Confusion eventually takes over! Here is a sample of the dialogue:
Costello: Look, all I wanna know is when you sign up the first baseman, how does he sign his name?
Costello: The guy.
Costello: How does he sign...
Abbott: That's how he signs it.
Abbott: Yes. PAUSE
Costello: All I'm trying to find out is what's the guy's name on first base. Abbott: No. What is on second base.
This confusion makes for entertaining misunderstanding. The audience who is listening eventually catches on and becomes an insider. They, therefore, become part of the joke.
Now as for your question of whether it makes for good humor - I would say that it does. There is something about misunderstanding that makes people laugh. From the ancient Greeks and Romans to today, people laugh at confusion.