Modus operandi :
1, Have the item you want remove the gem from in your inventory.
2, Have the Tainted Alkahest in you inventory as well.
3, Shift + left click on your item to pop up the window that shows you the stats+gems of the item.
4, Shift + right click the gem you want to remove and select the DELETE option.
N.B. : you do not have to use the tainted alkahest itself, the potion just disappears when you delete one gem, because yes, only one gem gets removed by the potion.