Hi all,
I read the mixed comments about the issue so I deliberately tried out claiming missions late.
I claimed an 1 hr mission 6hrs later and it worked fine, and also I claimed an 8hr mission after an overnight sleep and it also worked for me.
In both cases I wasn't online. (even the browser wasn't open)

The mission you do is listed at the very bottom of that window with the 'finish' button changing into a 'claim' button.

So I think the bug must come from somewhere else. (i was thinking on a mission reset sometime during the day, but that's not it, the other missions were swapped)
So as far as I know there is no need to force yourself to stay online (but that doesn't mean i discredit others bad experience, other bugs might cause you lose the progress)