What would happen to a teaspoon of neutron star material if released on Earth?

The reason that the density is so high is because the pressures are so immense. If we somehow teleported a teaspoonful of neutron star material to earth, it would very rapidly inflate because the pressures aren't high enough to crush it into its dense form. This would effectively be an enormous explosion.

It is difficult to describe what it would inflate out into - the neutron star material can be imagined as an incredibly dense soup of neutrons with some protons and leptons in small numbers. The protons and leptons would make neutron-rich elements like deuterium, but most of the matter would consist of free neutrons. These free neutrons would undergo beta decay to produce neutrinos, protons, and electrons, which would likely recombine to make very large amounts of hydrogen, some helium, and a few heavier atoms. In all of these cases, the atoms would be neutron-rich isotopes, though.

The behavior would look most like a very rapidly expanding gas. It would explode with such force that it wouldn't even need to "fall through the ground" - it would obliterate the floor entirely.

If we take neutron star material at say a density of $\sim 10^{17}$ kg/m$^{3}$ the neutrons have an internal kinetic energy density of $3 \times 10^{32}$ J/m$^{3}$. This is calculated by multiplying the number density of the neutrons $n_n$ by, $3p_{f}^2/10m_n$, the average KE per fermion in a non-relativistically degenerate gas and where $p_f =(3/8\pi)hn_n^{1/3}$ is the Fermi momentum.

So even in a teaspoonful (say 5ml), there is $1.5\times10^{27}$ J of kinetic energy (more than the Sun emits in a second, or a billion or so atom bombs) and this will be released instantaneously.

The energy is in the form of around $10^{38}$ neutrons travelling at around 0.1-0.2$c$. So roughly speaking it is like half the neutrons (about 250 million tonnes) travelling at 0.1$c$ ploughing into the Earth. If I have done my Maths right, that is roughly equivalent to a 40km radius near-earth asteroid hitting the Earth at 30 km/s.

So, falling through the Earth is not the issue - vapourising a significant chunk of it is.

Note that the beta decay of the free neutrons that dominate the neutron material is also energetic, but a slow process. On these 10 minute timescales, the neutrons could have exploded to a radius of a tenth of an au.