A bunch of my tasks keep getting moved to yesterday. Even if I select "Do it Today," they show up as being overdue when I refresh the page.
Please make sure both Todoist as well as your system and any devices you're using Todoist on are set to the correct time zone. If Todoist would ask you if you want to adjust your time zone, please agree.
Then, clear your browser's cache, log out and back in, and see if it helps.
The problem seems to have gone away on its own. Just to be clear, though, it was moving only some of my tasks. It had been working fine, and then suddenly about half of my tasks for today got moved to yesterday (and would move to yesterday again, immediately after I changed them to today), but the rest stayed put.