This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?
The Mayer-Vietoris sequence is an upshot of the relationship between sheaf cohomology and presheaf cohomology (a.k.a. Cech cohomology).

Let $X$ be a topological space (or any topos), $\mathcal U$ a covering of $X$. Let $\mathop{\rm Sh}X$ be the category of sheaves on $X$ and $\mathop{\rm PreSh}X$ the category of presheaves. The embedding $\mathop{\rm Sh}X \subseteq \mathop{\rm PreSh}X$ is left-exact; its derived functors send a sheaf $F$ into the presheaves $U \mapsto \mathrm H^i(U, F)$. For any presheaf $P$, one can define Cech cohomology $\mathrm {\check H}^i(\mathcal U, P)$ of $P$ by the usual formulas (this is often done only for sheaves, but scrutinizing the definition, one sees that the sheaf condition is never used). One shows that the $\mathrm {\check H}^i(\mathcal U, -)$ are the derived funtors of $\mathrm {\check H}^0(\mathcal U, -)$; and of course for a sheaf $F$, $\mathrm {\check H}^0(\mathcal U, F)$ coincides with $\mathrm H^0(\mathcal U, F)$. The Grothendieck spectral sequence of this composition, in the case of a covering with two elements, gives the Mayer--Vietoris sequence.

There is also a spectral sequence for finite closed covers, which is obtained as in anonymous's answer.

I guess that this can also be interpreted as Tilman does, in a different language (I am not a topologist).
