Are localized rings always flat as $R$-modules?

This is Proposition 2.1.16 in McConnel+Robson's book on Noncommutative Noetherian Rings.

As I mentioned in a prior post here there is a wealth of information on noncommutative localizations in Ranicki, A.(ed). Noncommutative localization in algebra and topology. ICMS 2002. In particular, there you will find an interesting paper on this very topic by Beachy: "On flatness and the Ore condition". Below is general reference information for flatness in the commutative case.

There is a very nice treatment of flatness in Bourbaki's "Commutative Algebra" - which begins with an excellent chapter on flat modules before turning to localizations in Chapter 2 (see Theorem 2.41. p. 68 for the result you seek). Also perhaps of interest is the following motivational remark from the introduction

The study of the passage from a ring $\rm A$ to a local ring $\rm A$, or to a completion $\rm \hat A$ brings to light a feature common to these two operations, the property of flatness of the $\rm A$-modules $\rm A$, and $\rm \hat A$, which allows amongst other things the use of tensor products of such $\rm A$-modules with arbitrary $\rm A$-modules somewhat similar to that of tensor products of vector spaces, that is, without all the precautions surrounding their use in the general case. The properties associated with this notion, which are also applicable to modules over non-commutative rings, are the object of study in Chapter I.

See also Atiyah and Macdonald, Corollary 3.6 and Proposition 3.10 pp. 40-41.