The question already has good answers but I think there is still more to be said.
References
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.
Sketch of a proof
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.
Etale descent
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.
[Edit]
Sketch of a proof, part 2
Let me give more details on the proof, now that I understand the details a little better. The main ingredients are
Nisnevich descent for perfect complexes, as a prestack of stable infinity-categories $X \mapsto Perf(X)$.
Compact generation of $D(X)$ by the perfect complexes, and also for the version "with support" on a closed subscheme $Z \subset X$, i.e. for the full subcategory $D_Z(X)$ of complexes that vanish on $X-Z$.
The Thomason-Neeman localization theorem: an exact sequence of stable infinity-categories induces a fibre sequence of K-theory spectra.
Then the proof is as follows. Let $j : U \hookrightarrow X$ be an open immersion and $p : Y \to X$ an etale morphism defining an elementary Nisnevich square
$$\require{AMScd}
\begin{CD}
W @>>> Y \\
@VVV @VV{p}V \\
U @>>{j}> X
\end{CD}$$
We want to show that the square
$$\require{AMScd}
\begin{CD}
K(X) @>{j^*}>> K(U) \\
@V{p^*}VV @VVV \\
K(Y) @>>> K(W)
\end{CD}$$
is a homotopy cartesian square of connective spectra. (In the noetherian case, Morel-Voevodsky showed that Nisnevich descent is equivalent to this Brown-Gersten-style excision property; in the non-noetherian case, this should really be taken as the "correct" definition of the Nisnevich topology.)
For this it is sufficient to show that there is an equivalence on homotopy fibres.
2) and 3) imply that the homotopy fibres are given by $K_{X-U}(X)$ and $K_{Y-W}(Y)$, respectively.
Then the equivalence follows from the fact that there is already an equivalence
$$ Perf_{X-U}(X) \stackrel{\sim}{\longrightarrow} Perf_{Y-W}(Y) $$
at the level of perfect complexes, by 1).
This proof works for quasi-compact quasi-separated schemes. By the way, the same proof works for (qcqs) derived schemes. By far the most non-trivial part of the proof is 2), which was established in [B. Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599]. A proof of 1), attributed to Drinfeld, is given in [D. Gaitsgory, Notes on geometric Langlands: Quasi-coherent sheaves on stacks, pdf]. In fact, this probably even gives a proof for the spectral schemes of Lurie, modulo a key point in the proof of 2) which I do not know how to do in the setting of $E_\infty$-ring spectra (but this is probably just my ignorance).
