What plastic-like materials allow IR to pass through them? What plastic-like materials allow IR to pass through them?
Material should be solid and plastic like. Does't care if it is transparent to light or not.
IR ray is emitted by Sharp distance center so it has to come back.
Don't know well about the length of the wave.
More about the sensor.
 Thanks for the help guys.
Some electronic devices' remote controls have their IR-diode covered with something. What is this material?
 A: If we look at you device (GP2Y0A51SK0F) specification file at Sharp website we find the following section (p. 6):

In case that protection is set in front of the emitter and detector portion, the protection cover which has the most efficient transmittance at the emitting wavelength range of LED for this product (λ=870nm±70nm), shall be
  recommended to use. The face and back of protection cover should be mirror polishing. Also, as there are cases that the characteristics may not be satisfied with according to the distance between the protection cover and this product or the thickness of the protection cover, please use this product after confirming the operation sufficiently in actual application.

So, we are talking about very near infrared. A good choice for the material would be PMMA — Poly(methyl methacrylate) (plexiglass), it is used for sensor covers in TV remotes for example. 
Here is an example IR transmission spectrum taken from here:

But you should check the plastic that you actually choose as it can contain additives.
A: Check this IR transmitting plastic. Its black and I use it for IR application sensors.
https://www.acrylite.co/extruded-black-9m020-gt.html
