The answer is always "no". By classification, a bielliptic surface over mathbbC has the form (EtimesF)/G where E,F are elliptic curves, G=subsetAut(E,0) is an abelian group acting by complex multiplications on E and by translations on F. (G is not necessarily cyclic as Tuan correctly points out.)
(X maps to an elliptic curve F/G and every fiber is isomorphic to an elliptic curve E, hence the name bielliptic.)
Then F acts on EtimesF by (x,y)mapsto(x,y+f), and this action commutes with the G-action. Thus, FsubsetAut0(X). As F is a projective variety, Aut0(X) is not affine.
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