Examples of failure of excision for homotopy groups ($\pi_k(X, A)$ is not $\pi_k(X/A, *)$)

Well, I figured some examples. The easiest is probably $S^2 \hookrightarrow S^3$ as an equator. Here $\pi_4(S^3, S^2) \cong \mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ by the exact sequence of a pair (the inclusion is obviously null-homotopic, so maps $\pi_k(S^2)\to\pi_k(S^3)$ are all zero). On the other hand, the quotient is a wedge of two three-spheres, and $\pi_4(S^3 \vee S^3)$ is equal to $\pi_4(S^3 \times S^3) = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ since $\pi_4$ depends only on 5-skeleton, while $S^3 \times S^3$ is $S^3\vee S^3$ with a 6-cell attached.

It is also true for $S^1\hookrightarrow S^2$, though you need more machinery. $\pi_3(S^2, S^1)$ is $\mathbb{Z}^2$ by the exact sequence of a pair as well. On the other hand, $\pi_3(S^2 \vee S^2) \cong \mathbb{Z}^3$ using the standard trick: fiber of $S^2 \vee S^2 \to K(\mathbb{Z}, 2)\times K(\mathbb{Z}, 2)$ has $H^3 \cong \mathbb{Z}^3$ by Serre's spectral sequence.