Carbon tax policy has been shown to be an effective incentive for the reduction of carbon emissions, and it also profoundly influences supply chain cooperation. This paper explores the interaction between carbon taxes and green supply chain cooperation. Specifically, we analyze the impact of a carbon tax on green supply chain coordination and further optimize the carbon tax to achieve a win-win situation for both the supply chain and the environment. Because consumer’s behavior has a significant impact on green product demand, we consider the problems above under two types of consumer’s behavior characteristics: consumer’s environmental awareness and consumer’s reference behavior. A game-theoretic model is employed to describe a green supply chain consisting of a manufacturer and a retailer, combining important factors such as the carbon tax rate, green investment coefficient, and degree of reference effect. Then, we obtain the optimal carbon tax rate by balancing the total tax revenue and product greenness. A revenue-sharing contract is introduced to achieve green supply chain coordination, and the impact of the carbon tax on coordination is analyzed. The results show the following. (1) The carbon tax rate and the difference between the power of the manufacturer and retailer are the main factors determining green supply chain coordination. (2) Maximum greenness can be achieved when development costs are higher, while the maximum tax revenue is obtained when the development cost is lower, but with the loss of greenness. (3) If the power of the manufacturer is low, coordination can be achieved under the optimal carbon tax. If the power of the manufacturer is at a medium level, coordination can be achieved by increasing the carbon tax; as a result, increased greenness will be realized, but with the loss of tax revenue. However, when the power of the manufacturer is strong, coordination cannot be achieved. (4) Price reference behavior can promote supply chain coordination, but consumer’s environmental awareness cannot.