To counter the outbreak of COVID-19, the accurate diagnosis of suspected cases plays a crucial role in timely quarantine, medical treatment, and preventing the spread of the pandemic. Considering the limited training cases and resources (e.g, time and budget), we propose a Multi-task Multi-slice Deep Learning System (M 3Lung-Sys) for multi-class lung pneumonia screening from CT imaging, which only consists of two 2D CNN networks, i.e., slice- and patient-level classification networks. The former aims to seek the feature representations from abundant CT slices instead of limited CT volumes, and for the overall pneumonia screening, the latter one could recover the temporal information by feature refinement and aggregation between different slices. In addition to distinguish COVID-19 from Healthy, H1N1, and CAP cases, our M 3Lung-Sys also be able to locate the areas of relevant lesions, without any pixel-level annotation. To further demonstrate the effectiveness of our model, we conduct extensive experiments on a chest CT imaging dataset with a total of 734 patients (251 healthy people, 245 COVID-19 patients, 105 H1N1 patients, and 133 CAP patients). The quantitative results with plenty of metrics indicate the superiority of our proposed model on both slice- and patient-level classification tasks. More importantly, the generated lesion location maps make our system interpretable and more valuable to clinicians.