Are there downsides to using resistors with too high power ratings?

The only usual downsides that come to mind are size, cost and sometimes resistors with a higher power rating use technologies such as wire wound that can have higher tolerance rating than some others. Other factors such as inductance that may be an issue with RF circuits won't be an issue for a current shunt.

So in general if you're happy with the size, cost and tolerance of the resistors you've found then there's no reason not to use a resistor with a higher power rating than you require.

Are there issues with using resistors that have power ratings much higher than required?

Bigger sized resistors may have higher self-inductance and this self-inductance in a component that is used to measure fast currents can lead to errors.

V across an inductor = L\$\dfrac{di}{dt}\$ and this is the problem.

Typically, a 3cm long piece of wire of 0.4mm diameter will have an inductance of 30nH (irrespective of its resistance) and if the current changes at 1 amp per microsecond (quite normal in a buck or boost converter), the volt drop due to di/dt will be 30mV. If the resistor is 3cm long and 0.03 Ohms, there will be a total volt drop of 60 mV for a sudden change of 1 amp flowing i.e. 50% bigger than what would be expected from a pure resistor.

Bigger sized resistors also have more self-capacitance (and capacitance to ground) and means another error when measuring high-speed signals. Here is a table from Vishay that shows tha parasitic inductance and capacitance of small SMT resistors - note the final columns: -

You can see that capacitance (parallel) and inductance (series) reduce with size and if you are operating above 1GHz resistor sizes become a big deal.

Also, as frequencies of RF transmissions rise there comes a point when the physical length of a resistor can cause standing wave problems. At 300MHz, RF has a wavelength of 1m and it is fairly unfeasible for anything significantly less than one-tenth of a meter will cause problems but what about wifi frequencies at 2.45 GHz - this has a wavelength of 12cm and conceivably, at this sort of frequency resistors that are about 1cm long will start to show problems.

WiFi is now occupying the 5 GHz band and some are starting to emerge at 60 GHz. 60 GHz has a wavelength of 5mm so realistically only 0201 resistors are not going to cause a significant problem.