Burning natural gas in any appliance produces byproducts that can potentially be harmful to people. This is the basic way that combusting fuel works—after all, a fireplace can be dangerous if the smoke from the fire isn’t allowed to vent up the chimney and instead gets pushed into the home.
This is a way of saying that, yes, a gas furnace can become dangerous the same way that combustion of any kind in a home can create dangers. That doesn’t mean that it’s common, or that you should feel afraid that your gas furnace is always on the verge of becoming a danger for your family. The modern gas furnace is built to extremely high standards and must pass numerous government requirements before one can be allowed into a house.