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Multi-view unsupervised feature selection has been proven to be efficient in reducing the dimensionality of multi-view unlabeled data with high dimensions. The previous methods assume all of the views are complete. However, in real applications, the multi-view data are often incomplete, i.e., some views of instances are missing, which will result in the failure of these methods. Besides, while the data arrive in form of streams, these existing methods will suffer the issues of high storage cost and expensive computation time. To address these issues, we propose an Incremental Incomplete Multi-view Unsupervised Feature Selection method (I$^2$MUFS) on incomplete multi-view streaming data. By jointly considering the consistent and complementary information across different views, I$^2$MUFS embeds the unsupervised feature selection into an extended weighted non-negative matrix factorization model, which can learn a consensus clustering indicator matrix and fuse different latent feature matrices with adaptive view weights. Furthermore, we introduce the incremental leaning mechanisms to develop an alternative iterative algorithm, where the feature selection matrix is incrementally updated, rather than recomputing on the entire updated data from scratch. A series of experiments are conducted to verify the effectiveness of the proposed method by comparing with several state-of-the-art methods. The experimental results demonstrate the effectiveness and efficiency of the proposed method in terms of the clustering metrics and the computational cost.