The remarkable performance of deep learning has sparked the rise of Deep Learning as a Service (DLaaS), allowing clients to send their personal data to service providers for model predictions. A persistent challenge in this context is safeguarding the privacy of clients' sensitive data. Oblivious inference allows the execution of neural networks on client inputs without revealing either the inputs or the outcomes to the service providers. In this paper, we propose FOBNN, a Fast Oblivious inference framework via Binarized Neural Networks. In FOBNN, through neural network binarization, we convert linear operations (e.g., convolutional and fully-connected operations) into eXclusive NORs (XNORs) and an Oblivious Bit Count (OBC) problem. For secure multiparty computation techniques, like garbled circuits or bitwise secret sharing, XNOR operations incur no communication cost, making the OBC problem the primary bottleneck for linear operations. To tackle this, we first propose the Bit Length Bounding (BLB) algorithm, which minimizes bit representation to decrease redundant computations. Subsequently, we develop the Layer-wise Bit Accumulation (LBA) algorithm, utilizing pure bit operations layer by layer to further boost performance. We also enhance the binarized neural network structure through link optimization and structure exploration. The former optimizes link connections given a network structure, while the latter explores optimal network structures under same secure computation costs. Our theoretical analysis reveals that the BLB algorithm outperforms the state-of-the-art OBC algorithm by a range of 17% to 55%, while the LBA exhibits an improvement of nearly 100%. Comprehensive proof-of-concept evaluation demonstrates that FOBNN outperforms prior art on popular benchmarks and shows effectiveness in emerging bioinformatics.