Automatic lesion segmentation on thoracic CT enables rapid quantitative analysis of lung involvement in COVID- 19 infections. Obtaining voxel-level annotations for training segmentation networks is prohibitively expensive. Therefore we propose a weakly-supervised segmentation method based on dense regression activation maps (dRAM). Most advanced weakly supervised segmentation approaches exploit class activation maps (CAMs) to localize objects generated from high-level semantic features at a coarse resolution. As a result, CAMs provide coarse outlines that do not align precisely with the object segmentations. Instead, we exploit dense features from a segmentation network to compute dense regression activation maps (dRAMs) for preserving local details. During training, dRAMs are pooled lobe-wise to regress the per-lobe lesion percentage. In such a way, the network achieves additional information regarding the lesion quantification in comparison with the classification approach. Furthermore, we refine dRAMs based on an attention module and dense conditional random field trained together with the main regression task. The refined dRAMs are served as the pseudo labels for training a final segmentation network. When evaluated on 69 CT scans, our method substantially improves the intersection over union from 0.335 in the CAM-based weakly supervised segmentation method to 0.495.