Those types of stoppers can be a real pain, I eventually just pulled mine out leaving the coverplate on the tub and use a plastic disc to cover the drain.
But, if I had to guess, I would guess that the metal plate that's on the back of the overflow cover is not catching the small arm that's supposed to keep the linkage up and prevent it from dropping down when the lever is up. Here's a link
that'll show the back of the overflow plate. You'll have to scroll down in the link to see the picture. You'll notice the small arm and how it catches on that metal plate. Yours could be either broken off or just old and worn down. You can buy new parts to make it work proper.