I have an air handler that serves both my hydroair heat and central AC. We always keep the thermostat switch to the "auto" position for the fan. We noticed the thermostat calling for heat but the air handler doesn't turn on. So we switch the fan switch to the "on" position and the fan blows and everything works. Then we switch it back to the "auto" position, it would not work again. Do you know what is the issue? How to fix it?

I have the same set up, and a similar (but slightly different) problem. When the thermostats are in auto mode, the heat works for a while, but eventually the handler won't run. I have to switch it to 'fan on' to get the blower to run. It's really frustrating, because on several occasions, I'll wake up in the morning, and there's no heat. I have to manually turn on the fan. If anyone has an answer to this problem, please help.

There are a couple of different ways to turn on the fan. The one I prefer is to have the fan operated through an aquastat attached to the hot water pipe coming out of the air handler.
Another is time based where in 'X' seconds after a call for heat the blower is powered.
Both involve a fan relay & since your fan will come on with the "fan on" thermostat setting, I have to believe the relay is the problem. In hydro-air systems the relay usually energizes the fan on a lower speed for heat than in cooling.