How fast can we measure using IR sensors
From a fundamental point of view there is practically no speed limit for IR detection (i.e., convert light intensity to an electrical signal). The frequency of the light is so high (in the 100 THz range) that there is practically no limit from that, given typical light modulation frequencies that can be electrically handled.

The fastest photodetectors for the IR that I know of, have cutoff frequencies in the 100 GHz range. This corresponds to rise times in the order of picoseconds. 

Well this is an extreme example. Practically you have to take into account other constraints, e.g., the size of the detector (smaller is faster) which has an impact on the field of view with optics applied, the cutoff wavelength (different detector materials have different "speeds"), the detector noise level (there are detector types that are very fast but not low noise), the desired amplification gain (the gain bandwidth product is usually somehow limited) or, the detector price.

Often the amplifier imposes the bandwidth limitation, and for the high speeds, an integrated module (i.e. detector and amplifier) is required.