What happens to a crop once it is filed as a loss with the insurance company?

If you can salvage any of it, you SHOULD notify the insurance company of the value of what you salvaged. Some people forget to do this though.  If there is no salvage value, plow it under and hope things go better next year.  The insurance company may want to send a claims adjuster to look at it though, so don't be too quick to do this if they haven't paid you yet.

It depends on what crop you're talking about, and what insurance company your policy is with.

An example is sprouted wheat. No good for making bread, but OK as a livestock feed. Two neighboring farmers could have sprouted wheat. If farmer A has a crop insurance policy with crop insurance company X, crop insurance company X's policy may stipulate that the crop be plowed under. Farmer B has a policy with company Y. Company Y may allow farmer B to sell his sprouted wheat for livestock feed (at a much lower price than if it were used for bread), and pay the farmer the difference.

Well before its filed as a loss it is inspected by an agent to determine the damage.  After the agent has established how much damage was done then you are issued a cash payment for the loss.