Let $X$ be a compact Kahler manifold, let $D$ be a smooth divisor in $X$, and let $U$ be a tubular neighbourhood of $D$ in $X$. Suppose that $D$ is Fano. Is it possible to extend every closed (1, 1)form on $D$ to a closed (1, 1)form on $U$?

Yes. For the proof, see e.g. http://arxiv.org/abs/math/0609617 Theorem 4.1. Here a stronger result is actually proven: Theorem: Let $(M, \omega)$ be a compact Kahler manifold, and $Z\subset M$ a closed complex submanifold. Denote by $[\omega]\in H^2(M)$ the Kahler class of $M$. Consider a Kahler form $\omega_0$ on $Z$ such that its Kahler class coinsides with the restriction $[\omega] Z$. Then there exists a Kahler form $\omega'$ on $M$ in the same Kahler class as $\omega$, such that $\omega Z=\omega_0$. An additional cohomological assumption is needed, because we build a global extension, and for an extension to a local neighbourhood you don't need it; the positivity (needed in assumption) is achieved by adding a big multiple of a Kaehler form. 

