Proving a category is *not* cartesian closed? Are there any general techniques for proving a category is not cartesian closed? Do cartesian closed categories have nice, easily checked properties that finitely bicomplete distributive categories typically do not have?
 A: A cartesian closed category means for every object $X$, $X\times -$ is a left adjoint, and thus preserves arbitrary colimits.  So typically what you would do is find some colimit which $X\times -$ does not preserve, maybe also by picking a clever choice of $X$.  Usually not too much cleverness is required.  Often some kind of "infinite" colimit works best.  I would say this is how failure of cartesian closure is usually noted.  I don't recall any examples where someone has actively sought after the failure of cartesian closure.
As the adjoint functor theorem suggests, if you can't find a colimit that $X\times -$ doesn't preserve for any $X$, then this is a strong hint that the category probably is cartesian closed, particularly when the category is cocomplete, though it could still fail for other reasons.  For locally presentable categories in particular, cocontinuity is equivalent to being a left adjoint.
If your category passes this battery of tests, you'll probably just need to use the definition of cartesian closure directly to show why it fails to be cartesian closed (if it does).  As a side note, the above applies to monoidal closure as well.
