Please read all of what I've written before taking any action.
Can you do us a favour and go through the phpBB changelogs / release notes to see if this is an actual bug that has been fixed? This would save me a lot of time. The forum presently runs phpBB 3.0.10. The latest available is 3.0.13-PL1. It would also be worthwhile googling to see if others have had the problem (you could also google for "Mark topics read" -- the overall functionality is the same).
Otherwise, if this is similar to past bugs that have been reported
(not this specific one, but others which are very similar in nature), the issue boils down to oddities with cookies or server-side sessions (we don't know which). The logging in/logging out situation -- ESPECIALLY if logging in from a different device, a different browser, or a different IP address
-- would explain how/why this is happening.