The answer to your question really depends on the nature of the reaction occurring. For example, if you have a chiral acid, HA* and an achiral base B, the product will be two molecules: *A- and HB+. However, if the reaction is a substitution, such as a chiral nucleophile, *A- reacting with an achiral alkyl halide, R-X:
*A- + R-Cl --> *A-R + Cl-
*A-R is a chiral molecule because it contains whatever chiral centers were present on *A-. If the other enantiomer of nucleophile were used, the other enantiomer of product would be obtained. If both enantiomers were used, both enantiomers would likely be obtained.
Overall, so long as your chiral center is not destroyed during the reaction, then the product will contain chiral centers and will thus be chiral or at least meso.