What am I failing to understand about a light in a material? In a material, a photon's velocity becomes slower, photon's wavelength becomes shorter, but photon's frequency doesn't change.
If there is a material that makes low frequency photons have very short wavelength photons, and if I stick my finger in the material, do I get a radiation burn?  What kind of effects can I have from the short wavelength photons in a material except for the refraction? 
 A: The energy of a photon is related to its frequency. In vacuum you can convert freely between frequency and wavelength - but once you have a material with a high refractive index, it's not so simple.
Your question about "radiation burn" is assuming that the energy of a short wavelength photon is always higher. That's not the case. The thing about the photon that doesn't change with medium is the frequency - and that is the thing you need to look at.
A: 
In a material, a photon's velocity becomes slower, photon's wavelength becomes shorter, but photon's frequency doesn't change.

The photon always travels with velocity c and its frequency and wavelength are fixed unless there are inelastic interactions.
The classical wave emerges from a confluence of photons, and it is the classical electromagnetic wave that goes with smaller than c velocity in transparent materials. Because of the way the classical em wave emerges from the constituent photons  one can qualitatively say that in a transparent material the photons scatter elastically and travel a longer path than the optical ray defining  the light they build up.
This makes no sense "If there is a material that makes low frequency photons have very short wavelength photons".
Low frequency has large wavelengths, period, by construction.
A light beam can give up all its energy, photons included,  to some solid, laser light can melt metals, and yes , your finger will be burned , but that has nothing to do with the original photons' frequency, except the delivery of so much energy that the metal can melt.
