Integral only control (as opposed to PI or PID)
Another way to look at your question is when would you use PI control with the P term 0.

The answer is basically "Whenever you think you can get away with it.".

This main risk with only integral control is oscillation or large overshoots due to windup.  If the output is low for a while, for example, then the integral term gets ever larger.  If this happens too quickly, the plant is driven harder and the output goes up before the controller has a chance to respond by lowering its output.

I-only controllers need to be well damped to avoid instability.  The plant needs to respond "quickly" in response to controller output before the integrator has a chance to wind up and cause unacceptable overshoot.  The equivalent is saying that the controller needs to be "slow" compared to the plant.

One advantage of I-only controllers is that the integral of the output is controlled long term.  For example, you might use something like this to control shaft speed, but where the total revolutions over some long time is more important than instantaneous shaft speed.

A real world example of such a requirement is the power line frequency.  It needs to be close to the nominal, but the total cycles over a longer term, like a day, are important to keep line-synchronous clocks correct.  The actual control of power line frequency is a lot more complicated than a I-only controller.  I'm only using this as a example of a control problem where the long term integral of the output is important, with a little noise allowed on the instantaneous value.