In my own (I use Ruida G6442 controller) I can set speed 0.1mm/s - so it's not possible directly to set speed to 0.
BUT I can use "fire" button on panel and beam will be firing as long I keep this or as was declared in setup.
Secondly, I can create multiple rounds circle (each over other one) with VERY tiny diameter - say 0.05mm which will be comparable to non-moving point.
Next - CO2 laser is not same as "weapon of destruction" shown in the movies and in practice you CAN'T cut way thicker and harder material just by setting speed to zero. Main reason (probably) is material thermal conductivity, vapors (absorption of energy NOT in the cutting area, but closer to beam source) and melted or already burnt material. So no, it's bad idea