This effect can be seen at the beginning of E1L3.MAP: "Death Row". Duke starts the level in an electric chair, and he has no weapons.

To make the player start weaponless, simply place a damaging floor texture under the start position. The only compatible damaging floor textures are FLOORSLIME (#200), HURTRAIL (#859), and FLOORPLASMA (#1082). PURPLELAVA (#4240) does not work with this effect.

If you'd like the player to start weaponless without taking any damage, lower the damaging floor a bit and place a blocked, floor-aligned sprite for the player to start on. Make sure the sprite is above the damaging floor (see the beginning of TUTORIAL.MAP for an example).