This post is for the benefit of anyone else experiencing this problem.
I noticed the Avast globe constantly spinning and the Avast service taking up a lot of CPU even when effectively nothing was running on the system. By looking at the Statistics page I noticed that Avast was repeatedly scanning taskinfo.exe, specifically Taskinfo 7. So I entered it into the exclusions list on the Avast settings page, and the problem is solved.
I can’t say why Avast had repeatedly scanned Taskinfo. Perhaps Taskinfo works by restarting itself every 1/10 second, and with each restart, Avast scans it again. In any case, just add it to the exclusions list.