Does zero integrality gap imply zero duality gap for certain problems?
Here is an instance that could be close to a counterexample to the
claim.

Consider the LP $P = \max \{ 1^T x \,|\, Ax \leq 1, x \leq 1, x \geq
0\}$ and its dual $P' = \min \{1^Ty + 1^Tz ~|~ A^Ty + z \geq 1, ~y
\geq 0, z \geq 0 \}$ for the $12 \times 6$ matrix

$ A =
\begin{bmatrix}
 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\
 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\
 1 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\
 0 &amp; 0 &amp; 1 &amp; 0 &amp; 1 &amp; 1\\
 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\
 1 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\
 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\
 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\
 1 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\
 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\
 0 &amp; 1 &amp; 1 &amp; 0 &amp; 0 &amp; 1\\
 0 &amp; 0 &amp; 1 &amp; 1 &amp; 0 &amp; 0
\end{bmatrix}\,.
$

An optimal solution of $P'$ is given by $y_1=y_2=y_{12}=1$ (all other
variables are zero), with the objective function value of $3$. The
Optimal solution of $P$ is given by the vector $x =
[0.5~0.5~0~1~0.5~0.5]^T$. If you solve $P$ as an integer program, the
optimal objective function value is only $2$, and $x = [1~0~0~1~0~0]$
is an optimal solution.

In summary, the LP $P'$ has an integral optimal solution, but its
dual, $P$ does not have an integral optimal solution. The primal-dual
roles are reversed from the set up that Ankur wanted. But given the
nature of LP duality, this instance could still be considered a
counterexample to the general statement of the original claim.