A global effort has been initiated to reduce the worldwide greenhouse gas (GHG) emissions, primarily carbon emissions, by half by 2030 and reach net-zero by 2050. The development of 6G must also be compliant with this goal. Unfortunately, developing a sustainable and net-zero emission systems to meet the users' fast growing demands on mobile services, especially smart services and applications, may be much more challenging than expected. Particularly, despite the energy efficiency improvement in both hardware and software designs, the overall energy consumption and carbon emission of mobile networks are still increasing at a tremendous speed. The growing penetration of resource-demanding AI algorithms and solutions further exacerbate this challenge. In this article, we identify the major emission sources and introduce an evaluation framework for analyzing the lifecycle of network AI implementations. A novel joint dynamic energy trading and task allocation optimization framework, called DETA, has been introduced to reduce the overall carbon emissions. We consider a federated edge intelligence-based network AI system as a case study to verify the effectiveness of our proposed solution. Experimental results based on a hardware prototype suggest that our proposed solution can reduce carbon emissions of network AI systems by up to 74.9%. Finally, open problems and future directions are discussed.