The electric flux due to a dipole is zero.But,the field due to the dipole is not zero.How?
If you have equal and opposite charges, the net flux is zero only for a surface defined to enclose the two charges.
If for instance, you decided to enclose one charge only with your surface which you are free to draw , you will have a net flux that is non zero.
Accordingly, zero flux does not mean zero field , but zero field means zero flux.