When we read "Inaugural Address of Harry S. Truman"

If you are reading the official title aloud, then you would read all of it.

Is "S." there read as /es/?

Yes. If you are being formal read it as "...Harry S. Truman"

.....and the 's' is pronounced /es/

If you are trying to be informal, you can leave it out.

You can find other examples of this, Americans love to informal with their Presidents. They refer to John F. Kennedy (formal) as : Jack Kennedy, John Kennedy and, very widely, as JFK.

Dwight David Eisenhower is hardly ever referred to be his full name. (That's what being famous, and President, will do for you. He's referred to as Eisenhower, everybody knows who you are talking about the President and not his son, David. Or you can just call him what he liked to be called : Ike. (That was his political campaign slogan "I like Ike".

I'm sure there are more examples, but I'm late for work.

Joe(be well)Nation

Leave it in if this is a title.