How is infra-vision any good?

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Let me address your second paragraph:

Upon thinking more about this, sight in the infrared band seems like it would come with several problems. This vision would be relying on the radiant emitted by blackbody radiation, according to their temperature. But, the frequency band of this emission relies on the temperature of the object, so two different objects at different temperatures should look....the same?

We have two key equations to consider: Wien's displacement law and the Stefan-Boltzmann law: $$\lambda_{\text{max}}=\frac{b}{T},\quad F=\varepsilon\sigma T^4$$ which give us the maximum wavelength at which an object emits radiation, and the flux from that object. Notice that $\lambda_{\text{max}}$ has a much weaker temperature dependence than $F$. As an example, say we have a box with one end at room temperature (293 K) and the other at the boiling point of water (373 K). The difference in $\lambda_{\text{max}}=2\mu\text{m}$ - a small enough difference given the size of the infrared band. I would argue that in most cases, your creatures could have an eye sensitive to objects with peak emission at wavelengths from $0.8\mu\text{m}$ to perhaps $8\mu\text{m}$, covering temperatures from a bit below the freezing point of water to several hundred degrees Fahrenheit.

Now, thermographic imaging works because hotter objects are more luminous - recall the Stefan-Boltzmann law. Even in a range of about $1\mu\text{m}$, the changes between the two ends will be drastic. The takeaway here is that yes, you will be able to distinguish objects at different temperatures if your infra-vision depends on the net flux from an object in the infrared band - in other words, if you use the thermographic imaging method.

What you need are proteins analogous to photopsins, the photoreceptors used by humans to detect visible light. The three photopsins act sort of like the filters in telescopes that allow astronomers to observe an object at select wavelengths (see photometric systems, such as the Johnson-Cousins filters). The three photopsins peak at about 420, 534 and 564 nm - too short for us.

However, related proteins have peak sensitivities at other bands - including non-visible bands. The protein phytochrome peaks at far-red wavelengths (750-800 nm); it's not too far out of the question to imagine a protein peaking at $\sim1\mu\text{m}$. Indeed, research on some species of fish has shown sensitivity in the near-infrared portion of the spectrum (and the sensitivity has indeed been traced back to the eyes), indicating that opsins of some sort are indeed active at wavelengths of $\sim0.8\text{-}0.9\mu\text{m}$.

In short, we want a protein with a sensitivity curve roughly like the purple one below. This is a modified version of a figure based on Fig. 1 of Bowmaker & Dartnall 1980. I've extended the axis and added in a qualitative example of how the opsin sensitivity should behave in the $0.7\text{-}0.8\mu\text{m}$ range:

^{Original image by Wikipedia user Maxim Razin under the Creative Commons Attribution-Share Alike 3.0 Unported license. Image modified to include qualitative behavior of the hypothetical opsin protein.}