Does the thickness of germanium dioxide affect the amount of infrared light that passes through it?

Furthermore, what properties of germanium cause it to behave in this manner?

The thickness of any material will affect the amount of IR radiation that passes through it. The equation for the attenuation of radiation thru matter is given by this equation. $I = I_0\ e^{-kt}$ where k is the attenuation coefficient for radiation through the material and t is the thickness of the material. 
Germanium dioxide is transparent to IR radiation but that doesn't mean 100% of IR passes through a thin layer of GeO2. There will be a small amount that gets attenuated. The equation above holds but k will be very small. Still, the thickness will determine how much IR passes through it.