If $C_*$ is exact sequence of free $R$-modules, and $F:_R\text{Mod}\to _S\text{Mod}$ is additive functor, $FC_*$ is in general not exact. But if $R$ is a field, is $FC_*$ is exact?
(I think since $C_*$ broken into free short exact sequences, and these short exact sequences splits, so they are taken by $F$ to exact sequences.)
Yes: as you say, $F$ is an exact functor since additive functors take split short exact sequences to split short exact sequences. But an exact functor preserves exact sequences of any length. For this latter statement, see for instance here: Equivalence between exactness of additive functors .