In the bad old days of just four years ago it was pretty common for mobile users to get shunted off to some half-baked, feature-deprived "mobile" version of the website they were trying to visit. This misguided practice was common (and annoying) enough that even today Chrome for Android and other mobile web browsers ship with a feature that allows users to "request desktop site."
To make that feature work Chrome for Android changes its user agent string. Any site that uses user agent strings to redirect mobile users will no longer because to redirect them and the desktop version is displayed.
Responsive websites don't rely on user agent strings though. Instead they adapt to screen size based on CSS media queries so even if a user has the option for desktop sites checked in Chrome they still won't get the "desktop" site (of course with responsive sites there really is no desktop site, just a desktop layout).
Provided your responsive designs are good, this isn't a problem (and if they aren't then you have bigger problems). However, Opera web standards evangelist Bruce Lawson raises an interesting edge case: what about users that have never seen the mobile layout and are disoriented when they do? If you were expecting, say, the desktop layout of the BostonGlobe.com and instead saw the mobile layout for the first time you might be understandably confused. Here's what Lawson has to say:
The first thing to note is that this isn't a problem unique to responsive sites. The same thing would crop up with a separate mobile experience. The difference is the inability to opt out of the responsive layout. An edge case? Sure, but Lawson isn't alone in wondering about turning off responsive designs. CSS guru Chris Coyier tackled that very question last year, writing:
I would agree with both points, but clearly there are at least a few edge cases where offering an option to turn off responsive design might be a good idea. Of course it may not be worth worrying about the edge case of unfamiliar visitors – that's the sort of decision you can only really make by looking at your own visitors and doing your own testing.
If you actually want to try it, Coyier has some ideas on how to go about creating an option to opt out of a responsive design.