If you could connect the North and South poles via a tube and you fell into one of the openings, would you come out the other side?
Heat and pressure from magma does not apply. Thinking on the basis of a falling body and the affects of speed and gravity as it passes through the Earth.
Assuming you're speaking of geographic North and South Poles (or any other two points on the planet that connect by going through the center of the sphere) you would fall into the tube, accelerating towards the center of the Earth. Once you reached the center, the force of gravity pulling you "downward" would be zero, but your momentum would speed you past the center at a maximum velocity. Then, you would begin to decelerate, slowing down as you reached the other end. At the other opening, your velocity would be zero, and if you don't scramble out of the hole, you would fall back in towards the center, and would continue oscillating until you slowed down and stopped at the center. Then it would be as if you were floating in space, as all the Earth is pulling at you from all sides.
Of course this is a purely theoretical imagination of mine.
I assume that the earth is pure spherical. So the tunnel from NP to SP has to pass through the spherical centre. And the tunnel is diametrical. When you are vertically sinking (perpendicular to the polar horizontal plane), the gravitational force on you are of from both upper and lower part of the earth. At a postion at a h kilometer down (or deep ) from the NP, you are pulled by the gravitational force from above and below part of the earth. Of course the partial gravitional force gets cancelled as you are pulled equally by both sides as you are through the diameter, which devides earth symmetrically from all sides parallel to the horizontal plane. So it is vertical force which has greater pull from below than the vertical pull from above till you reach the centre. The centre of the earth is ideal place where gravitational force of pull is equal from all sides on you. You may feel that from all the directions (from up, from below,from left, from right , and from whatever angle you think in an equal to that but from an opposite angle) you are beeing equally pulled there. And of course the net force on you is zero. By the property inertia and the velocity you gained will continue to cross the geocentre with hightest velocity. Now you move from the centre towards SP, till you loose all the velocity and kinetic energy gained and reach up the SP gaining the potential energy. But if you are not boarding at SP, then you will again go back after completing the diametriacal amplitude and continue your adventurous great simple harmonic motion. In this type of SHM, the force on you is maximum when you are at extremities (i.e, north or south poles) and zero the mean postion at the centre of earth and your velocity is zero at the poles and it is highest at the centre of the earth.
Hope this helps.