The question already has good answers but I think there is still more to be said.
As already mentioned, algebraic K-theory satisfies Zariski descent. For regular noetherian schemes this is due to
- Kenneth S. Brown, Stephen M. Gersten, Algebraic K-theory as generalized sheaf cohomology, Higher K-Theories, Lecture Notes in Mathematics Volume 341, 1973, pp 266-292.
It was generalized to finite dimensional noetherian schemes by
- R. Thomason, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, vol. III, Progress in Mathematics, vol. 88, Birkhäuser, Basel, 1990, pp. 247–435.
However, there is a stronger statement: it satisfies descent with respect to the Nisnevich topology (which lies between Zariski and etale). This is due to
- Yevsey A. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, Algebraic K-theory: connections with geometry and topology, 1989, pp 241-341.
and was generalized to finite dimensional noetherian schemes in the same paper of Thomason.
In the following paper the above results are extended to finite dimensional quasi-compact quasi-separated schemes.
- Andreas Rosenschon, P.A. Ostvær, Descent for K-theories, Journal of Pure and Applied Algebra 206, 2006, pp 141–152.
I learned from Peter Scholze that in modern language a proof can be given, after identifying Quillen K-theory with the K-theory of the stable infinity-category of perfect complexes, by using the following characterization of extendibility of perfect complexes: a perfect complex on an open subscheme $U \subset X$ can be extended up to quasi-isomorphism to a perfect complex on $X$, if and only if its class in $K_0(U)$ lies in the image of $K_0(X)$. This gives a homotopy fibre sequence of connective spectra $K(X \text{ on } Z) \to K(X) \to K(U)$ where $Z \subset X$ is the closed complement of an open subscheme $U \subset X$. I am not sure in exactly what generality this proof works. The significance of working with infinity-categories is that the presheaf of infinity-categories $U \mapsto \mathrm{Perf}(U)$ satisfies descent, unlike its triangulated shadow.
Finally, let me mention that the question of etale descent is closely related to the Lichtenbaum-Quillen conjecture. This is now a theorem of Rost and Voevodsky and it implies that K-theory does satisfy etale descent in sufficient large degrees. The theorem of Trobaugh that Steven Landsburg mentioned, about etale descent for mod-$\ell$ Bousfield-localized K-theory, is in
- R. Thomason, Algebraic K-theory and étale cohomology, Ann. Sci. Ecole Norm. Sup. 18 (4), 1985, pp. 437–552.
and is also generalized in the paper of Rosenschon and Ostvær.