There could be a few things keeping your lifter from...well...lifting, so we will start with the easy then move down from there
Lifter not plugged in
Very simple solution here and I trust you will figure it out (Hint: plug it in). It will fit into the middle AUX port in the back of the headset base.
Headset base in DHSG
Lifter will not lift when headset base is in DHSG. Move to RHL (underneath the charging base) and it should start lifting, if not...
Note: You MAY need to do a resync after moving the DHSG switch. If, when you turn the headset on again after moving it, there is no sound, then perform a resync.
Wall mount tab installed
Some phones are designed to be mounted on a wall, and to hold the handset to the base, there will be a little tab. Works great for when the phone is vertical, but gets in the way for lifters. It can usually be removed by sliding it out. Easy peasy.
Shoulder rest installed on phone:
Sometimes those big slug looking things installed on handsets will weigh them down, making it too heavy for the lifter to lift. If possible, try either removing it or grabbing a handset that doesn't have that piece installed.
If none of these are the problem, then try our other methods (unplugged reset, resync, etc...) Still not lifting, it may need to be replaced.