This issue was caused by having a ~5 MB TIFF file with several hundred pages on the desktop next to the picture I was viewing. When viewing an image, Windows Photo Viewer apparently does some kind of preloading of files in the same directory in the background.
In the case of this multi-paged TIFF file, each page was apparently transformed into a large temporary file in the temp directory while the viewer was open, quickly using up all remaining disk space. Deleting the TIFF file resolved the issue.