Is there a buiding code that requires a lavatory undermount sink to have an overflow hole?

Plumber is telling me that without this vanity overflow, it is code violation and therefore when time comes to sell my house, I need to get the sink replaced.

I have never heard of a code violation for this application. You will have to have him prove to you that there a code.

Whether kitchen undermount or bathroom undermount, they are making all kinds of sinks with no overflow hole, so how can it be a problem with code ?

No, I would have to say no because I can't find anything that says there is a code for that.

Edit: Loretta, enter 'undermount sinks' in Google and you will see sample sinks, kitchen or bathroom, with no overflow hole.

Tick

Tickle nailed it.
Plumber is telling me that without this vanity overflow, it is code violation
Ask your plumber if he has ever seen a vessel sink lavatory? They don't have overflows and more and more undermount bowls are being manufactured without a overflow.
Regards, Tom