How does one study Krull dimension of some Iadic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull dimension of the ring, but what can we say in general case?

For a Noetherian ring R, the Krull dimension of its $I$adic completion, $\hat{R}$ is given by $\sup h(J)$, where $J$ ranges over all maximal ideals of $R$ containing $I$ and $h(J)$ is the height of $J$. Therefore $\dim \hat R\le \dim R$ with equality only when $I\subset \operatorname{rad} R$. A reference is "Topics in $\mathfrak m$adic topologies" by S.Greco, P.Salmon 

