People have definitely done it, particularly for opening cabinet doors or opening doors over a projector in a home theater set up. However, I would be concerned about the amount of humidity in that environment. Most people who build automatic toilet flushers are pushing the handle on the outside of the tank, or even better, a button on top of the toilet. Those are pretty easy to build. (I myself am quadriparetic, use a power wheelchair and have limited use of my hands, so this actually is a topic that I’ve discussed before. No laughing. 
)
Anyway, if you want to see the cabinet door projects, just remember that they are all working with the old architecture, but the physical device would be the same. I think firgelli (mentioned in the following thread) would probably be the best place to start.
But again, I don’t know about the environment. You could talk to the company, there are probably some built for boats that might handle the humidity.
Again, if you go with a button push toilet, you have a lot more options, including just a regular battery-operated button pusher.
Switchbot makes a battery-powered button pusher that works with SmartThings. But you definitely don’t want to put that inside the toilet tank.
(You should be able to use a linear actuator rather than a circular servo, because unlike blinds, you don’t need a bi directional device. You only actuate a toilet flush control in one direction.)
Tagging @johnconstantelo and @ogiewon in case they have any additional thoughts.