The assumption imply that the natural embedding induces an isomorphism $\Gamma(X,\mathscr O_X)\simeq \Gamma(\overline X,\mathscr O_{\overline X})$. Then this means that  the complement of $X$ has at least codimension $2$ and if $X$ is normal, or at least $S_2$, then the same condition holds for every *reflexive* sheaf.

So, **if** $\Omega_{\overline X}$ is a reflexive sheaf, then the restriction 
$\Gamma(\overline X,\Omega_{\overline X})\to \Gamma(X,\Omega_{\overline X})=\Gamma(X,\Omega_{X})$ is an isomorphism.

**EDIT** removed intro paragraph about the starting assumption.