In edge inference, an edge server provides remote-inference services to edge devices. This requires the edge devices to upload high-dimensional features of data samples over resource-constrained wireless channels, which creates a communication bottleneck. The conventional solution of feature pruning requires that the device has access to the inference model, which is unavailable in the current scenario of split inference. To address this issue, we propose the progressive feature transmission (ProgressFTX) protocol, which minimizes the overhead by progressively transmitting features until a target confidence level is reached. The optimal control policy of the protocol to accelerate inference is derived and it comprises two key operations. The first is importance-aware feature selection at the server, for which it is shown to be optimal to select the most important features, characterized by the largest discriminant gains of the corresponding feature dimensions. The second is transmission-termination control by the server for which the optimal policy is shown to exhibit a threshold structure. Specifically, the transmission is stopped when the incremental uncertainty reduction by further feature transmission is outweighed by its communication cost. The indices of the selected features and transmission decision are fed back to the device in each slot. The optimal policy is first derived for the tractable case of linear classification and then extended to the more complex case of classification using a convolutional neural network. Both Gaussian and fading channels are considered. Experimental results are obtained for both a statistical data model and a real dataset. It is seen that ProgressFTX can substantially reduce the communication latency compared to conventional feature pruning and random feature transmission.