This is applicable to both the weapon palette (F5) and the clothing/unit palette (F6). You can unequip weapons and units, but not clothing!
To unequip an item, select the slot you wish to clear by clicking on the weapon or unit icon itself (not the number or the gear button below it). This will bring up a window which displays a number of items that can be put in that slot. To unequip the item rather than replace it with anything, click the button at the bottom of this window, indicated in the below screenshot.