Shortest Equivalent CNF Formula
The problem of determining whether there is an equivalent CNF formula with at most a given number of literals is $\Pi_2^p$-complete.  The version minimizing the number of clauses is within a factor of $n$ of the formula size, where $n$ is the number of variables.  See section 6 of:


Christopher Umans, The Minimum Equivalent DNF Problem and Shortest Implicants, JCSS 63(4), 597–611, 2001.  doi:10.1006/jcss.2001.1775 (preprint)


A recent result shows that computing a particular lower bound for the size of the shortest equivalent CNF formula (measured by number of clauses, as you specify) is NP-complete.  This paper also states that your problem of minimizing the number of clauses is $\Pi^p_2$-complete as well, citing the Umans paper above, although why this follows is not immediately obvious to me.


Ondřej Čepek, Petr Kučera and Petr Savický, Boolean functions with a simple certificate for CNF complexity, DAM 160(4–5), 365–382, 2012.  doi:10.1016/j.dam.2011.05.013