Can schlieren photography be done with a mobile phone camera?

Yes. Schlieren photography does not require a professional camera specifically, and it can be done even with an older or lower-quality phone camera. The camera is not the critical part of the setup.

What matters most is the optical arrangement. A schlieren setup typically needs a spherical mirror (a parabolic mirror also works), plus a light source and proper alignment. One answer notes that a mirror from a cheap Newtonian telescope can be used. The phone camera can then be placed where the observer or camera would view the image in a knife-edge/Foucault-style arrangement.

So the limiting factor is not megapixels or whether the camera is a phone or DSLR, but whether you have the required mirror and can align the system correctly. A better camera may make capturing results easier, but the schlieren effect itself comes from the optics, not from using a “professional” camera.