Do you Open damper when using gas fireplace?

No, you do not need to open the damper.

Example - A gas cook stove is not vented and neither is the oven. Nor were the old gas space heaters (I still use them) your grandparents used. Carbon monoxide is produced by incomplete burning of fuel. Not just the burning of fuel. In fact the EPA states that a correctly operating gas appliance DOES NOT produce unsafe levels of carbon monoxide. If they did, all our grandparents would be dead. I remember sleeping by an old gas space heater turned on high.

Ventless gas logs are sold world wide, if just the burning of the gas produced CM then they would not sell logs that allowed you to close your damper. The bigger concern with these logs is not CM but oxygen depletion. All vent free logs have an oxygen depletion safety on them and not a CM safety. That should tell you something. So relax. If you are concerned, go to Home Depot and buy a CM detector. I think every home with gas appliances should have one in case an appliance gets faulty.