Interpreting results of a thermal camera - what should I expect to be hot?
I know that heat is caused by current


Not really, no.

Temperature rises because of electrical power getting converted to heat. The amount of power converted determines the temperature rise, not the current alone. For example, a current of 0.1 A through a 1 MΩ resistor will cause 10 kW of heating – whereas a current of 10 A through a 1 Ω resistor will only lead to 0.1 kW of heating.

But that's only half of the story – heat alone doesn't mean a temperature increase; what counts is how much a specific material gets heated by a specific amount of heat power (i.e. a 1 kg piece of wood needs less heat to get 1°C hotter than 1 kg of water), and how hard it is for the component to get rid of the heat (i.e. you add a large heatsink, the temperature doesn't rise as much).

So, any component that wastes a lot of electrical power AND has a hard time getting rid of the heat is going to get hot. 


  A small amplifier chip I'm using has a very hot resistor (compared to the rest of the circuit) as soon as power is on, and eventually the main IC and pieces of the PCB nearby heat up too. Is this sort of a case-by-case basis where it depends on my circuit diagram, or are there general rules of thumb when looking at thermal imaging of circuits?


That resistor probably intentionally drops some voltage e.g. to supply the amplifier chip.

Yes, it's a case-by-case basis, because the same device (e.g. an amplifier) can be built with architectures of different efficiency, i.e. how much of the input power is converted to heat vs converted to output power. Generally, power electronics deal with lots of power, and thus, even that 1% waste heat of a 99% efficient device means a lot of heat.