Would a black hole created on the surface burrow through the crust? If scientists created a microscopic black hole with an initial mass of one ton on the surface of the earth, would the gravitational attraction to the center be enough for it to "burrow" until it eats its way through the crust? It seems like there would be a bad outcome. How dense would rock have to be to form a barrier?
 A: Black holes this small will have very high Hawking temperature:
$$
T_H = \frac{\hbar c^3}{8 \pi G M k_B} \approx 10^{20}\,\text{K},
$$
So, before this black hole can fall down even the diameter of an atom it will evaporate through Hawking radiation. As a result, the 1 tonne of black hole mass would be converted into the energy of very high energy particles of all kinds.
Some of these particles, such as neutrinos, gravitons and weakly interacting particles (if they exist), will fly away without interacting with anything on Earth, others would interact, resulting in the formation of the giant fireball analogous to hydrogen bomb explosion. The yield will be hundreds times larger than the Tsar Bomba explosion, so potentially it can wipe out the considerable part of any continent, but most likely will not end life on Earth.
A: As dmckee said in his comment, the black hole would fall towards the center of the Earth.
To specifically answer this portion of your question:

How dense would rock have to be to form a barrier?

There is absolutely no density of rock or anything else that would stop or even slow it down.  Even if you created this microscopic black hole on the surface of a neutron star, it would fall right through to the center.
It would oscillate back and forth through the center of the Earth, almost certainly evaporating faster than it consumed material.  The evaporation rate would eventually be high enough that the Hawking radiation coming off of it would be extremely damaging to surrounding material.  There is a lot of energy in a one ton microscopic black hole!
I wonder though... There may be a density of material high enough (like a neutron star) that the microscopic black hole would consume material at a fast enough rate that instead of evaporating, it would grow and completely consume the surrounding planet / neutron star.  I would be interested in seeing a calculation that tries to answer that!
