This paper investigates task-oriented communication for edge inference, where a low-end edge device transmits the extracted feature vector of a local data sample to a powerful edge server for processing. It is critical to encode the data into an informative and compact representation for low-latency inference given the limited bandwidth. We propose a learning-based communication scheme that jointly optimizes feature extraction, source coding, and channel coding in a task-oriented manner, i.e., targeting the downstream inference task rather than data reconstruction. Specifically, we leverage an information bottleneck (IB) framework to formalize a rate-distortion tradeoff between the informativeness of the encoded feature and the inference performance. As the IB optimization is computationally prohibitive for the high-dimensional data, we adopt a variational approximation, namely the variational information bottleneck (VIB), to build a tractable upper bound. To reduce the communication overhead, we leverage a sparsity-inducing distribution as the variational prior for the VIB framework to sparsify the encoded feature vector. Furthermore, considering dynamic channel conditions in practical communication systems, we propose a variable-length feature encoding scheme based on dynamic neural networks to adaptively adjust the activated dimensions of the encoded feature to different channel conditions. Extensive experiments evidence that the proposed task-oriented communication system achieves a better rate-distortion tradeoff than baseline methods and significantly reduces the feature transmission latency in dynamic channel conditions.