There is an option for this. Press the "ESC" button and go to the "game" tab. Next, uncheck "Show damage effects on HUD". After, you shouldn't have the effects anymore.
[EDIT: You can't disable certain effects such as hitting an explosion, nitronome effects, etc.]
There were a lot of complaints about the red screen effects and shaking that came with the pet update. You can disable it now. I don't think there is any way to disable all shakes, a Nitronome blast will always make your screen do a backflip.