Privacy is of worldwide concern regarding activities and processes that include sensitive data. For this reason, many countries and territories have been recently approving regulations controlling the extent to which organizations may exploit data provided by people. Artificial intelligence areas, such as machine learning and natural language processing, have already successfully employed privacy-preserving mechanisms in order to safeguard data privacy in a vast number of applications. Information retrieval (IR) is likewise prone to privacy threats, such as attacks and unintended disclosures of documents and search history, which may cripple the security of users and be penalized by data protection laws. This work aims at highlighting and discussing open challenges for privacy in the recent literature of IR, focusing on tasks featuring user-generated text data. Our contribution is threefold: firstly, we present an overview of privacy threats to IR tasks; secondly, we discuss applicable privacy-preserving mechanisms which may be employed in solutions to restrain privacy hazards; finally, we bring insights on the tradeoffs between privacy preservation and utility performance for IR tasks.