"AI as a Service" (AIaaS) is a rapidly growing market, offering various plug-and-play AI services and tools. AIaaS enables its customers (users) - who may lack the expertise, data, and/or resources to develop their own systems - to easily build and integrate AI capabilities into their applications. Yet, it is known that AI systems can encapsulate biases and inequalities that can have societal impact. This paper argues that the context-sensitive nature of fairness is often incompatible with AIaaS' 'one-size-fits-all' approach, leading to issues and tensions. Specifically, we review and systematise the AIaaS space by proposing a taxonomy of AI services based on the levels of autonomy afforded to the user. We then critically examine the different categories of AIaaS, outlining how these services can lead to biases or be otherwise harmful in the context of end-user applications. In doing so, we seek to draw research attention to the challenges of this emerging area.