I'll tackle this question in two parts:

1. What are "fish lungs" or their equivalent?
Lungs are specialized organs that allow humans to breathe - that is exchange more oxygen-rich air from outside the body with the oxygen-poor and carbon dioxide rich air in our lungs (collected from the body and concentrated in the lungs). This is termed "gas exchange" and is a fundamental principle of respiration/breathing. Many fish (not all as a few utilize a similar version to humans) have something called "gills" that acts like lungs for fish, and they often look like an ampitheater of red seats, and are thin layers on top of each other (in this analogy, these seats are blood vessels that allow for the flow of oxygen/carbon dioxide). Also, unlike our lungs, gills are not active in drawing air into them; instead as water moves over the gills, oxygen in the water goes into the gills, while carbon dioxide in the gills goes into the water. They tend to be exposed to the water, sticking a bit out of the fish, but have some protection from larger objects, and this will play into #2.

2. Can these collapse/be destroyed?
Now with some background out of the way, can fish "lungs"/gills collapse? There are two broad scenarios where this may occur: one, damage to the gills in water; and two, damage to the gills when the fish end up out of water. For the former scenario, if a large enough object hits or predator attacks the gills, which are often exposed to the water, the gills may collapse like tissue paper being crumbled, causing the fish to be unable to breathe correctly. When the fish is exposed to dry air, the gills are unable to hold their structure, and again crumble and stick together, collapsing and preventing the fish from breathing correctly.

For related information and another interesting read, look at: ScienceLine.

No and yes. No because fish do not have lungs; they have gills instead. Gills fulfill the same function as lungs, extracting oxygen from the surrounding fluid (which, for fish, is water rather than air), but they are on the outside and depend on the water to remain open. This leads to the answer of "yes": when taken out of water, the gills are no longer supported and therefore collapse/close. When in this state, the gills cannot be used to extract oxygen because the fluid (now air, not water) cannot reach the parts of the gills where oxygen is absorbed.

There is more information in the ScienceLine answers here. [There are some fish with adaptations to breathe air, such as the air-breathing catfish. These still use gills rather than lungs though, and their gills collapse shut when the fish are removed from water.]

They can if the fish is heavy enough to not be strong enough to keep its lungs open. I don't know to what extent this is a problem, though.

Fish have gills for getting oxygen under water. The gills collapse if the fish is out of water.
ScienceLine.

The first note is to mention that fish do not have lungs (generally speaking), but rather they have gills. Like lungs, gills are designed to adsorb oxygen from the surrounding environment and transport it to the blood of the animal. Gills are specifically designed to do this in water, while lungs are designed to do this in air.

Now about how gills can collapse- The more surface area the gills have, the more effective they are at adsorbing oxygen (because more water will be in contact with the gill). To create larger surface area, gills have many thin wrinkles that float when suspend in water. When in air, these wrinkles stick together, effectively collapsing the gills.

Fish do not have lungs, they have evolved a way of "breathing" that is different from ours since we are in the air and they are in the water. A fish gulps water into it's mouth and passes it through the gills. Oxygen in the water is absorbed by the gills into the bloodstream.

In human, the lung is usually attached to the chest wall. When air or liquid enters this space, the lung deflate and therefore collapses.

Lungfish is the only group of fish with lungs. Unlike human lungs, when lungfish exhale, the lungs collapses due to compression from the water. In fact, lungfish lung secretes certain chemical to prevent the lung surface from adhering when they collapse.