In the article "Tannaka duality for geometric stacks" (arxiv, see nlab for a summary) Jacob Lurie introduced the notion of a tame abelian tensor category. An abelian tensor category is called tame if every short exact sequence $0 \to M' \to M \to M'' \to 0$ with $M''$ flat remains exact after tensoring with an arbitrary object. Basically this means that flat objects behave in the usual way. For example it is easy to check that in a tame abelian tensor category every extension of flat objects is flat (Lemma 5.4).
Question 1: Do you know any further literature about tame abelian tensor categories?
Question 2: Can you give an explicit example of an abelian tensor category which is not tame? Jacob Lurie remarks that every abelian tensor category with enough flat objects is tame, because then we can define $Tor_1$ and use symmetry. So in particular, a counterexample does not have enough flat objects. [Tom Goodwillie has given an example below]
Question 3: Can you give an example of a cocontinuous tensor functor between tame abelian tensor categories which is not tame?