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 kind of thing is discussed in

- Bhargav Bhatt,
*Algebraization and Tannaka duality*, 2014, arXiv.

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.