On the backside of that overflow cover there is a little steel spring that holds the lever in the open position. Quite often as they age the springs will rust away. The good news is, it is a very simple and inexpensive repair.

Remove the two screws that hold the overflow trim cover in place, then pull upward & outward on the trim cover at the same time. You will see a wire rod linkage that is connected on the inner end of the lever and it goes down to the stopper. Continue pulling upwards and the stopper will come out, or at least get the stopper high enough so that when you remvove the trim cover you can let the linkage hand down inside the tub, otherwise the linkage might fall back down the overflow pipe.

There is a little brass cotter pin that passes through the linkage and lever. Remove the cotter pin. The trim cover will now be disconnected. You might be able to bend the spring a bit to restore the proper action but generally it is much easier to just go to your local hardware and get a new trim cover for a couple bucks.