The Internet produces a continuous stream of new documents and user-generated queries. These naturally change over time based on events in the world and the evolution of language. Neural retrieval models that were trained once on a fixed set of query-document pairs will quickly start misrepresenting newly-created content and queries, leading to less effective retrieval. Traditional statistical sparse retrieval can update collection statistics to reflect these changes in the use of language in documents and queries. In contrast, continued fine-tuning of the language model underlying neural retrieval approaches such as DPR and ColBERT creates incompatibility with previously-encoded documents. Re-encoding and re-indexing all previously-processed documents can be costly. In this work, we explore updating a neural dual encoder retrieval model without reprocessing past documents in the stream. We propose MURR, a model updating strategy with regularized replay, to ensure the model can still faithfully search existing documents without reprocessing, while continuing to update the model for the latest topics. In our simulated streaming environments, we show that fine-tuning models using MURR leads to more effective and more consistent retrieval results than other strategies as the stream of documents and queries progresses.