Ok, I'll state this a bit more confidently... (but I'm still worried I'm missing something).

Any time we're in, say, a stable model category where every object is cofibrant (or stable $\infty$-category), then given an object $X$ and a homotopy push-pull diagram (which I won't draw), involving $A \rightarrow B$, $A \rightarrow C$ and $C, B \rightarrow D$ we get a long exact sequence like
$$
\cdots \rightarrow [D, X] \rightarrow [B, X]\oplus [C, X] \rightarrow [A, X] \rightarrow [\Sigma^{-1} D, X] \rightarrow\cdots
$$
Indeed, we have a homotopy push-pull as above precisely if we have a (co)fiber sequence
$$
A \rightarrow B \oplus C \rightarrow D
$$
where the first map is the difference of the two obvious ones. There's probably a good way to do this without being fancy, but the easiest way I see to do this is in the setting of $\infty$-categories: the pushout and cofiber displayed above both have manifestly the same universal property. (I think I'm using cofibrancy here to say that the coproduct and homotopy coproduct should agree... maybe I don't need this- I'm bad with model categories, someone should correct me.)

Anyway, this *must* be in one of the obvious references. Adams? Neeman's book on triangulated categories? Something like that.

It's not immediately obvious to me that this specializes to Mayer-Vietoris, but that's me revealing too much of my ignorance. Inclusions of open subsets don't seem to be cofibrations, so why should the usual square we write down be a homotopy pullback/pushout (after taking suspension spectra)?