Question

What would happen if a rocket ran out of fuel in the Earth's atmosphere?

Answer 1

It would do exactly what a rock or a frisbee does when you toss it.

After the engines cut off, it couldn't get any more energy from
anywhere, and after that, as it pushed air aside to get through,
and had air molecules scraping against it, those would slowly
rob kinetic energy from it.  Sooner or later it would run out of
kinetic energy, start falling, and it would eventually make either
a big 'SPLOOSH' or else a big 'CRUNCH', depending on exactly
where it returned to Earth's surface.

Answer 2

The rocket would fall back down to the surface of earth because there would be nothing propelling it upwards.