lol you guys are nuts

I just want to flush the toilet by moving my hand across the IR beam
and the servo push the handle down.

If I used a car solenoid it would break the handle of the toilet
lol

and I don't want a whole bunch of messy wires just something plan and simple.

If I put wires up connecting from the door to the toilet there just going to take it down when we have guests.

lol

No one has a solution

electronically? lol

I am surprised you guys haven't said to attach a cap to the door knob and if they don't flush the toilet then IR receiver sends the signal to power the transistor to charge the cap, and shock them when they touch the handle.
lol

Good Idea though

I just want something clean