Ionizing Radiation by the Doppler Effect Would a sufficiently fast relative velocity between an observer and a light source, which radiates non-ionizing light (form its perspective) make said light become ionizing from the observer's perspective? 
 A: Yes. If the source is approaching the observer (or equivalently the observer approaching the source), the radiation it is emitting gets blue-shifted. However, it would have to be moving very close to the speed of light for i.e. a microwave emitter for the observer to see gamma rays.
The formula is f = f0 sqrt((1+v/c)/(1-v/c)), where positive v is approaching (blueshift). The frequency goes to infinity as v approaches c. For ultra-relativistic (near-light-speed) velocities it is approximately f = 2 y, with y = 1/sqrt(1-v^2/c^2) being the Lorentz factor.
There are actually several examples:
Synchrotron light: Electrons moving near the speed of light pass by wiggler magnets and the acceleration causes them to radiate. The electrons get wiggled with GHz (microwave) frequencies. However, Lorentz contraction (by a factor of y=100000) causes the electrons to be wiggled at optical frequencies in their own frame, so an observer moving with the electron would see it glow with light in the ~infared range. However, the electron's light is blue-shifted again by 200000 and formed into a beam, creating the brightest lab xray source known.
Greisen–Zatsepin–Kuzmin limit: Some cosmic rays, usually protons, are moving so fast that the y ~ 10^11 and the cosmic microwave background, in the particles frame, becomes a beam of gamma rays pushing back on the particle. These rays can interact with the particle by creating high energy particles, slowing it down and providing an energy limit to cosmic rays.
A: @YuvalWeissler Energy is always dependent on the reference frame. As a simple Classic example, consider the kinetic energy of an object as measured in different reference frames in which it has a different velocity: the same car can produce a greater damage (work) if you see it approaching your own from your front or from your back! Energy (alone) is neither a relativistic invariant. 
Finally, the question was mentioning a "relative" velocity" - there is no way to consider the source at rest and the observer in motion, as there's no way to choose a privileged reference frame.
