This study explores the use of generative AI for automating the classification of tutors' Dialogue Acts (DAs), aiming to reduce the time and effort required by traditional manual coding. This case study uses the open-source CIMA corpus, in which tutors' responses are pre-annotated into four DA categories. Both GPT-3.5-turbo and GPT-4 models were tested using tailored prompts. Results show that GPT-4 achieved 80% accuracy, a weighted F1-score of 0.81, and a Cohen's Kappa of 0.74, surpassing baseline performance and indicating substantial agreement with human annotations. These findings suggest that generative AI has strong potential to provide an efficient and accessible approach to DA classification, with meaningful implications for educational dialogue analysis. The study also highlights the importance of task-specific label definitions and contextual information in enhancing the quality of automated annotation. Finally, it underscores the ethical considerations associated with the use of generative AI and the need for responsible and transparent research practices. The script of this research is publicly available at https://github.com/liqunhe27/Generative-AI-for-educational-dialogue-act-tagging.