It may leave some soot on the outdrive, and be a bit louder, but not having it won't damage anything. However. Part of the exhaust system is the flapper you have on the outdrive. It helps prevent water from washing up the outdrive and flooding the exhaust with water. I would invest in a new one.
If your talking about the flap on the snout?
It's needed. It stops the water from being sucked up into the motor when you slow down.
Possible to suck up a bunch O water and do some damage.
