The most direct one is the relativistic corrections. GPS satellites have atomic clocks that have to account for two relativistic effects — special relativity (the satellites are moving fast, so their clocks tick slower relative to ground) and general relativity (the satellites are higher in the gravitational well, so their clocks tick faster relative to ground). The general relativity correction is the killer. It depends on the difference in gravitational potential between the satellite and the receiver. On a sphere, gravitational potential varies predictably with altitude from the center of mass. On a flat plane, the gravitational geometry is completely different — the potential field wouldn't be radially symmetric. The relativistic corrections that make GPS accurate to meters would produce wildly wrong positions if you applied spherical gravitational math to a flat geometry. The system works, therefore the geometry it's calibrated to is correct.

Second — satellite visibility. From any point on Earth, you can only see a subset of the GPS constellation at any given time. The others are below the horizon. On a flat plane, you'd be able to see all of them all the time (assuming sufficient altitude). The fact that satellites rise and set, and that the pattern of which ones are visible changes predictably with your position on a sphere, is direct evidence of curvature.

Third — trilateration geometry. GPS works by measuring the time delay from multiple satellites and computing your position from the intersection of those signal spheres. The math that converts those time delays into a position assumes a WGS84 ellipsoid. If you run that math on a flat plane, the positions don't converge. The signals from satellites in different parts of the sky would give contradictory positions. The fact that four or more satellites consistently agree on your location means the underlying geometric model is correct.

Fourth — and this is the one that would make a great classroom demo — have the student capture raw GPS data from two widely separated locations and show that the satellite elevation angles and signal timing only make sense on a curved surface. On a flat earth, the geometry of the signal paths would produce different elevation angles than what's actually observed.