Neural passage retrieval is a new and promising approach in open retrieval question answering. In this work, we stress-test the Dense Passage Retriever (DPR) -- a state-of-the-art (SOTA) open domain neural retrieval model -- on closed and specialized target domains such as COVID-19, and find that it lags behind standard BM25 in this important real-world setting. To make DPR more robust under domain shift, we explore its fine-tuning with synthetic training examples, which we generate from unlabeled target domain text using a text-to-text generator. In our experiments, this noisy but fully automated target domain supervision gives DPR a sizable advantage over BM25 in out-of-domain settings, making it a more viable model in practice. Finally, an ensemble of BM25 and our improved DPR model yields the best results, further pushing the SOTA for open retrieval QA on multiple out-of-domain test sets.