If all objects emit IR radiation, would it be sufficient to place an IR mirror behind an object to propel it forward? I'm assuming objects don't move all the time because they emit IR radiation in all directions. If we have an IR mirror behind the object to focus their direction, would this propel the object forward?
 A: Yes, this would work, but the amount of thrust created would be pitifully small.  For example, a sphere of radius 1 cm at 300 K radiates
$$\sigma T^4 \cdot A = \left(5.67 \times 10^{-8} \,\mathrm{W \cdot m^{-2} \cdot K^{-4}}\right) \cdot \left(300 \,\mathrm{K}\right)^4 \cdot \left(4\pi (1 \,\mathrm{cm})^2\right) = 0.577 \,\mathrm{W}$$
where $\sigma$ is the Stefan-Boltzmann constant, $T$ is the temperature, and $A$ is the surface area.  This is enough power to produce a thrust of $(0.577 \,\mathrm{W}) / c = 1.93 \times 10^{-8} \,\mathrm{N}$.  That's a measurable effect, but it's not very impressive.
Still, even a very small amount of thrust, applied consistently over a long period of time, can eventually make an object go pretty fast.  What would it take to turn this device into a viable rocket engine?  Well, you'd need to increase the temperature a lot and add a powerful, long-lasting heat source.  That prompts the idea of the nuclear photonic rocket – essentially, a very hot nuclear reactor propelled by black body radiation.  Some people think this could be an (at least somewhat) plausible idea for space flight outside of the solar system.
A: Your mirror idea will work, as it says in the answer by Thorondor, but the amount of force is very small. Indeed this is reminiscent of the way an ordinary rocket motor works. In the rocket case the motor emits matter as well as light, and there is a 'mirror' (i.e. the wall of chamber) at the back of the motor so that the hot matter is all ejected in one direction, thus resulting in thrust (momentum transfer to the body of the rocket). If the materials could withstand the temperatures, then at high enough temperature you could get a substantial contribution from the momentum of the electromagnetic radiation. It would be thermal radiation, but not in the infra-red part of the spectrum because the temperatures are so high. It would have to be gamma rays.
Here is a related and well known phenomenon. If you take an ordinary object, say metallic or ceramic, and make one side of it rough and black, and the other side smooth and silver, then the thermal radiation will be emitted more by the black side than the silver side, resulting in a small net radiation pressure force. In outer space (i.e. in vacuum) this would be enough to cause a little acceleration. As long as it is hot enough, the object will accelerate in the direction of the silvered side (the side with reduced emission).
You can buy a small device called Crookes radiometer intended to demonstrate this phenomenon with vanes rotating in an evacuated glass bulb, but be warned: the devices on the market almost never get a good enough vacuum to make the radiation pressure effect the main contribution. In an imperfect vacuum there is a further effect from gas molecules colliding with the vanes. In this case the device still functions as a radiometer (a way of measuring electromagnetic radiation) but it would not be a demonstrator of radiation pressure.
A: No, it would not work. The mirror  reflects thermal radiation in the direction of the object and this gives radiation pressure. However it intercepts the radiation that would otherwise come through its location so nothing changes. This assumes that the object is in equilibrium with an isothermal environment so there will be isotropic thermal radiation. If this assumption is not met the object will accelerate due to anisotropic irradiation and emission even without a mirror. In such cases a mirror will have an impact.
