Did Richard Gere actually do the tap dance routine in the movie
Chicago, where the announcer said, "Ladies & Gentlemen, a tap dance."

According to the following two interviews, Richard Gere does the tap
dancing in the movie Chicago. He mentions that he had a very good tap
dancing teacher. However it doesn't say if he is actually doing the
routine that you mentioned in your question.

BBC Interview with Richard Gere
http://www.bbc.co.uk/films/2002/12/11/richard_gere_chicago_interview.shtml

BBC NewsOnline Entertainment 
http://news.bbc.co.uk/1/hi/entertainment/film/2556603.stm

Given what Gere has said in interviews, he probably did his own tap
dancing...However, even a bad dancer can look good on film: Nothing
need be shot straight-through, and it can be edited according to the
dancer's ability. This is not to say Gere is a bad dancer! It may be
that he had experience dancing in the past, since he says he used to
do musicals before he became a movie star. :)

Kriswrite

Grudy-ga--

I'm a TV producer and recently interviewed Rob Marshall, the director
of "Chicago."  I asked him about Gere's tapping and about how many
journalists and critics are skeptical he did his own dancing because
you never see his face.
Marshall said he was "outraged" that people felt that way.  That
indeed Gere did all his own dancing, as did Zellweger and Zeta-Jones.
(although personally, I'm suspicious that R & C they did their own
cartwheels).  And because people are so distrustful, Marshall put on
the end credits of the movie, "Richard Gere's singing and tap dancing
are performed by Richard Gere."  You'll see the same thing for Renee &
Catherine.   As a former tapper, and I do believe Gere did his own
dancing -- but he was learning short sequences of steps.  He had never
tapped before and in only 4-6 months it's doubtful he could sustain a
long, elegant dance.  But by having him do little snippets and editing
it in fragments, voila!  He's a dancer!

It's probably rather like Humphrey Bogart's fighting.

He was hopeless!

But, by shooting a piece at a time, he was made to look an adept.