Speech technology has been increasingly deployed in various areas of daily life including sensitive domains such as healthcare and law enforcement. For these technologies to be effective, they must work reliably for all users while preserving individual privacy. Although tradeoffs between privacy and utility, as well as fairness and utility, have been extensively researched, the specific interplay between privacy and fairness in speech processing remains underexplored. This review and position paper offers an overview of emerging privacy-fairness tradeoffs throughout the entire machine learning lifecycle for speech processing. By drawing on well-established frameworks on fairness and privacy, we examine existing biases and sources of privacy harm that coexist during the development of speech processing models. We then highlight how corresponding privacy-enhancing technologies have the potential to inadvertently increase these biases and how bias mitigation strategies may conversely reduce privacy. By raising open questions, we advocate for a comprehensive evaluation of privacy-fairness tradeoffs for speech technology and the development of privacy-enhancing and fairness-aware algorithms in this domain.