How to drop a dwarf, or how to drop a dwarf? If you want to drop any random dwarf - indeed, several in quick succession - build a U shape with walls. In the dip of the U, add a 1x1 retracting bridge. Beside this bridge, place a connected pressure plate. Designate the internal wall for deconstruction. It should look something like...WWWW
C+P
WWWW
Walls; C bait wall, designated for removal; + retracting drawbridge; P linked pressure plate.
This will cause someone, usually a child, to go remove the wall, and in the process will flip the plate, and then fling themselves off the bridge, down a 2-3 level fall that you should have already prepared. That dwarf will wander to the hospital, and someone else will come to remove the building immediately, until your entire fort is injured. Be careful of this, and remove the designation and/or block the entry when you've injured enough dwarves.
If you want to injure a specific dwarf, then you should build a long bridge at the end of a narrow hallway, with a lever on each end:
WWWWWWWWW
WL+++++L
WWWWWWWWW
Link both levers to the bridge. The right lever should be open for anyone. The left lever is profiled for your one dwarf. Have that dwarf pull the lever, and they walk back into the main hall. The bridge, on a short delay, will disappear once he's walked some, allowing him to drop. Then order the right lever pulled a few times, so that you can reset the trap from the outside without having to deconstruct anything (or put pressure plates at the bottom of the pit, as they'll trigger once the victim falls). This has the advantage that it targets a specific dwarf, so you can toss your lye maker in, but is slower and a bit more micromanagement.