= Solution
The complement $Y$ is covered by the $m$ principal affine opens $D(f_1),\ldots,D(f_m)$. Every finite intersection is again a principal affine open, so this is an acyclic cover for $\mathcal O_Y$. Its <Čech cochain complex> has no degree-$m$ term because there is no intersection of $m+1$ distinct cover members. Therefore
$$
\boxed{H^m(Y,\mathcal O_Y)=0.}
$$
Translate $p$ to the origin. The complement $\mathbb A_k^3\setminus\{0\}$ has the affine cover $D(x),D(y),D(z)$, and its second Čech cohomology is
$$
\frac{k[x^{\pm1},y^{\pm1},z^{\pm1}]}
{k[x^{\pm1},y^{\pm1},z]+k[x^{\pm1},y,z^{\pm1}]+k[x,y^{\pm1},z^{\pm1}]}.
$$
The class of $x^{-1}y^{-1}z^{-1}$ is nonzero, so $H^2(\mathbb A_k^3\setminus\{p\},\mathcal O)\ne0$. On the other hand, $\mathbb A_k^3\setminus\ell=D(x)\cup D(y)$, and the first part with $m=2$ gives $H^2(\mathbb A_k^3\setminus\ell,\mathcal O)=0$. Since <sheaf cohomology> is invariant under scheme isomorphism, the two complements are not isomorphic.
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