I would recommend looking at a perfectly viable example that's been done in the real world: snakes. Many snakes (pitvipers, for instance, or rattlesnakes) have special pit organs in their heads, which for your purposes can be considered a set of eyes that view the far infrared spectrum instead of the visual spectrum.
Operates in the infrared band
Obviously, this is covered, although it needs to be emphasized that eyes seeing the visual spectrum are incapable of also seeing into the far infrared spectrum: the necessary structure required is too different. No, this doesn't rule out seeing the visual spectrum as well, provided you have eyes together with those pit organs: those pitvipers I mentioned have more ordinary eyes as well, and they work just fine.
Incidentally, the snakes in question have a thermal sensitivity estimated at >0.001C, so even minor temperature differences are easily detectable. Rattlesnakes actually use this vision to aim for their prey's weak points.
If you are walking down a curved passageway of uniform, smooth stone, you must be able to distinguish the curve in the passageway
I believe the result will be comparable to walking down a city street at night, with the lights out: dark, yes, perhaps difficult to see at a distance, but not outright indistinguishable. You'll have minor thermal fluctuations in the stone, enough to give notice that something is there. Besides, people tend to be pretty good about not walking into walls even without having to touch them.
If you are standing in a cavern with a lake in it, you must be able to distinguish the surface of the lake from a particularly smooth patch of floor.
At a distance, probably not, unless the lake is home to life of some sort. At close range, though, you should be readily able to separate the two before you actively step in the water.
If you are hunting from underground mushrooms, you must be able to distinguish by sight between Ripened Feast mushroom, which tastes like cheddar cheese, and a Pustulent Death mushroom, which liquifies your skin. These mushrooms smell exactly alike, and have the same exact shape, and can only be distinguished by their 'color'.
This isn't fully answerable without a more detailed description of the differences. However, if the two mushrooms in question maintain different temperatures (for instance, if one tends to 22C and the other 18C), this would be trivial.
Now, it is important to note a significant caveat to far-infrared vision. Snakes cannot see very far with this vision, and even then their visual resolution is poor. If you don't want badly nearsighted cave people with fuzzy vision, the pit organs will need to be larger than in snakes (larger than human eyes, in other words, if you want anywhere near the same precision), and/or possibly more than two of those sensory organs.
Cultural interpretation of how your people might work with such organs is an exercise for the reader, but I suspect that blows to the face will be taken much more seriously (as posing a risk for blindness, even in the absence of active attempts at eye-gouging), probably with some societal taboo against touching another's face without permission, much less in the way of facial cosmetics, etc. I wish you the best of luck in figuring out other potential implications (read: further worldbuilding, possible points of interest from which stories can be launched, etc.).