Greenwashing refers to practices by corporations or governments that intentionally mislead the public about their environmental impact. This paper provides a comprehensive and methodologically grounded survey of natural language processing (NLP) approaches for detecting greenwashing in textual data, with a focus on corporate climate communication. Rather than treating greenwashing as a single, monolithic task, we examine the set of NLP problems, also known as climate NLP tasks, that researchers have used to approximate it, ranging from climate topic detection to the identification of deceptive communication patterns. Our focus is on the methodological foundations of these approaches: how tasks are formulated, how datasets are constructed, and how model evaluation influences reliability. Our review reveals a fragmented landscape: several subtasks now exhibit near-perfect performance under controlled settings, yet tasks involving ambiguity, subjectivity, or reasoning remain challenging. Crucially, no dataset of verified greenwashing cases currently exists. We argue that advancing automated greenwashing detection requires principled NLP methodologies that combine reliable data annotations with interpretable model design. Future work should leverage third-party judgments, such as verified media reports or regulatory records, to mitigate annotation subjectivity and legal risk, and adopt decomposed pipelines that support human oversight, traceable reasoning, and efficient model design.