What word means "taking something to unrealistic extremes"?

It sounds like you're looking for synonyms of unlikely. If so, implausible, far-fetched, or outlandish might suit you. In this context, dubious would also fit.

If you're looking for a phrase, someone might say "That's a reach" or "I don't see that happening any time soon."

Person B: With the amount of traffic you get, your server should be fine.

Person A: If a billion people view my web site at once, my server will crash.

Person B: That is a contrived example.

The example Person A has given is made up simply to find a flaw in the observation Person B has made, so it is contrived

adjective - obviously planned or forced; artificial; strained: a contrived story.

Equally you could call the argument put forward by Person A a straw man argument

A straw man is a component of an argument and is an informal fallacy based on misrepresentation of an opponent's position

In your example the idea of a billion people all visiting the site at once is a straw man because it won't ever happen.