Does current flowing through a zener diode, when forward biased, increase due to heavy doping?
This depends entirely on the particular diode.
All diodes have a design tolerance for not only reverse bias, which is specialised with zener diodes as they are able to conduct once the breakdown voltage is reached, but also forward bias. If the forward current exceeds the design tolerance a diode can be destroyed just like any other component. In the same way, if the reverse bias voltage and/or current design limitations are exceeded then zener diodes will be destroyed just like regular diodes.
In general, a forward biased zener diode will perform the same as a regular diode. It will have a voltage drop across the PN junction of about 0.7 V. The data sheet for the particular diode, whether it is a zener diode or 'ordinary' diode, will provide specific details on how much current it is able to conduct (in either direction).