The title says it all.  

A very similar question was asked and answered about linear groups, but none of the counterexamples are algebraic:
http://mathoverflow.net/questions/22814/are-extensions-of-linear-groups-linear

If $A$, $B$ are affine and there is a rational section of $C \to A$ in $1 \to B \to C \to A \to 1$, then $C \to A$ is affine, so $C$ is affine.  But if not?