Question: It's clear that one shouldn't use floating precision when working with, say, monetary amounts since the variation in precision leads to inaccuracies when doing calculations with that amount.

That said, what are use cases when that is acceptable? And, what are the general principles one should have in mind when deciding?

Answer: 
I'm guessing you mean "floating point" here. The answer is, basically, any time the quantities involved are approximate, measured, rather than precise; any time the quantities involved are larger than can be conveniently represented precisely on the underlying machine; any time the need for computational speed overwhelms exact precision; and any time the appropriate precision can be maintained without other complexities.

For more details of this, you really need to read a numerical analysis book.
It's appropriate to use floating point types when dealing with scientific or statistical calculations. These will invariably only have, say, 3-8 significant digits of accuracy.

As to whether to use single or double precision floating point types, this depends on your need for accuracy and how many significant digits you need. Typically though people just end up using doubles unless they have a good reason not to.

For example if you measure distance or weight or any physical quantity like that the number you come up with isn't exact: it has a certain number of significant digits based on the accuracy of your instruments and your measurements.

For calculations involving anything like this, floating point numbers are appropriate.

Also, if you're dealing with irrational numbers floating point types are appropriate (and really your only choice) eg linear algebra where you deal with square roots a lot.

Money is different because you typically need to be exact and every digit is significant.