If gravity disappeared, would Newton's third law make everything that was pressed to the ground by gravity get pushed upwards?

As other answers explain, Newton's third law wouldn't push you upwards, because reaction disappears as soon of action (gravity) vanishes.

However, we need to keep in mind that we are siting on several thousand kilometres of rock heavily compressed by its own weight. If weight suddenly disappears, that rock will react like a spring and project itself and anything in the surface at very high speed to space. In fact, even the most conservative ballpark estimates of the elastic deformation of Earth in its present state are in the order of several kilometres, so that's the quite instant rebound we can expect.

Yes, but in almost all cases the push would be imperceptible.

Reaction forces from surfaces occur when the molecules in the wall are displaced from their equilibrium position. The harder they are pushed the more they are displaced, and the more they are displaced the harder they pushed back. When you stand on a surface without falling it is because you have displaced the surface enough for the reaction force to match your weight.

As an extreme example of this imagine standing on a trampoline. Heavier people make the trampoline surface sink lower than lighter people. The same is true on hard surfaces, but the displacement is basically imperceptible.

If you suddenly removed gravity, the reaction force from the displaced surface would still be there, and it would push you away until the equilibrium state of the surface is restored.

Again, imagine standing on a trampoline holding heavy weights. When you throw the weights away the trampoline will begin to push you up until you reach a new equilibrium. If the weights were heavy enough it could even launch you into the air.

The same thing would happen harder surfaces, but the amount of time the remaining reaction force would act on you would be tiny, and you would notices hardly any effect.

You can simulate this experiment in real life with an electromagnet.

For instance, you can hold a vertically oriented steel plate by a horizontally oriented electromagnet, so that the plate is free to fall down, when the electromagnet is deenergized.

If the falling plate has a horizontal velocity component and describes a parabola, you can conclude that the normal force has pushed it. If the plate falls straight down, you can conclude that the there was no push.

Even without performing such experiment, you, probably, can predict that the plate will fall straight down. This is because the normal force is a reaction force and it never exceeds the applied force causing it, be that electromagnet attraction or gravity. So, as the applied force disappears, gradually or suddenly, the normal force will disappear with it and, therefore, there won't be any push.