I have a pretty new house with a new furnace. I have a pretty sophisticated home control thermostat made by control4 and I've gone through a couple sets of backup thermostat batteries lately (1/month). When the batteries die, the furnace turns on and bakes the space.

Is this likely a bad transformer or some other problem?

For a new and sophisticated system you should not need batteries to run or even be in the T-stat to help it work. The batteries will not help the system work if the power is off. If you have a bad transformer you have no heat,cooling or blower at any time but your thermostat will have a display. I would say its a bad thermostat.

I need to add "except for backup when power fails."

Before replacing the thermostat, make sure it has a good connection to the common if it needs one.