Homotopical perspective on the long exact sequence in homology and Mayer-Vietoris

Let $f : A \to X$ be a based map of based spaces. The homotopy pushout $X \coprod_A \text{pt}$ is called the homotopy cofiber, cofiber, or mapping cone of $f$; I'll denote it by $X/A$. Iterating this construction produces the cofiber sequence or Puppe sequence

$$A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma X/A \to \dots$$

which is in some sense the ancestor of all long exact sequences for relative homology and cohomology, although it's easier at this point to describe how to get the long exact sequence for relative cohomology. If $Z$ is another based space, then taking spaces of maps into $Z$ turns the cofiber sequence into a fiber sequence

$$[A, Z] \leftarrow [X, Z] \leftarrow [X/A, Z] \leftarrow [A, \Omega Z] \leftarrow [X, \Omega Z] \leftarrow [X/A, \Omega Z] \leftarrow \dots$$

which is built out of taking homotopy pullbacks in the same way that the cofiber sequence is built out of taking homotopy pushouts. If $Z$ is an Eilenberg-MacLane space $B^n G = K(G, n)$, taking $\pi_0$ of this fiber sequence produces the long exact sequence in relative (reduced) cohomology up to degree $n$, and taking $n \to \infty$ and piecing together the results gives the whole thing.

This is discussed a little in May's Concise Course in Algebraic Topology and a lot in Strom's Modern Classical Homotopy Theory.

I haven't seen the Mayer–Vietoris sequence in the answers here, but it can be seen as a special example of the cofiber sequence in the accepted answer.

Given a pointed excisive triad $(X;U,V)$, replace $X$ with the double mapping cylinder (homotopy pushout) $X'$ of the inclusions of $W = U ∩ V​$ in $U$ and $V$, with the interval through the basepoint collapsed to a point. Then the cofiber sequence starts

$$U \vee V \longrightarrow X' \longrightarrow \Sigma W \longrightarrow \Sigma U \vee \Sigma V.$$

It's a nice exercise to check the last map can be seen as the difference of the suspensions of the inclusions from $W$ to $U$ and $V$. Applying $H^*$ (or some other cohomology theory), the first two maps turn into ring maps (so products of coboundaries are zero), and the last one a map of graded groups (actually, of $H^*X$-modules). Its desuspension is the standard "difference of restrictions" map $H^*U \times H^*V \to H^* W$.