COVID-19 pneumonia is a disease that causes an existential health crisis in many people by directly affecting and damaging lung cells. The segmentation of infected areas from computed tomography (CT) images can be used to assist and provide useful information for COVID-19 diagnosis. Although several deep learning-based segmentation methods have been proposed for COVID-19 segmentation and have achieved state-of-the-art results, the segmentation accuracy is still not high enough (approximately 85%) due to the variations of COVID-19 infected areas (such as shape and size variations) and the similarities between COVID-19 and non-COVID-infected areas. To improve the segmentation accuracy of COVID-19 infected areas, we propose an interactive attention refinement network (Attention RefNet). The interactive attention refinement network can be connected with any segmentation network and trained with the segmentation network in an end-to-end fashion. We propose a skip connection attention module to improve the important features in both segmentation and refinement networks and a seed point module to enhance the important seeds (positions) for interactive refinement. The effectiveness of the proposed method was demonstrated on public datasets (COVID-19CTSeg and MICCAI) and our private multicenter dataset. The segmentation accuracy was improved to more than 90%. We also confirmed the generalizability of the proposed network on our multicenter dataset. The proposed method can still achieve high segmentation accuracy.