Should I care about IC input pin current?
If the input voltage you provide to the IC inputs lies between its rails, then the IC will draw only the current it needs.

Where an IC may take excessive current is when the input voltage goes outside its rails, and input-to-rail diodes turn on. These diodes are there to protect against static electricity (ESD), but only have a limited current handling, usually in the low 10s of mA. A common way to do this if there is the possibility of an input going beyond the rails is the use of a series resistor. Obviously this needs to be sized to not interfere with the normal operation of the input.

There are cases where you might think you are keeping within the rails, but turn out not to be. A fast edge on a long mismatched line may overshoot at an input sufficiently to make an ESD diode conduct.  The effect of this can range from nothing bad, to a momentary upset of function, to full-blown damage of the input. Usually a small series resistor at the transmitting end of the line, in the low 10s of ohms, will mitigate this.

I'm guessing that what you mean by 'sink current' is the current that needs to be pushed into/out of the pin by the device driving it. This was of the order of a mA in the bad old TTL days, but these days CMOS inputs mean the static current is essentially negligible. Be warned that some inputs may use internal pullups/pulldowns so that they don't need to be connected for default operation. The data sheet should give an explicit current value that is needed for correct operation.

Although static current is negligible, inputs can have capacitance in the many pFs, sometimes even 10s of pFs, and a large dynamic current may be needed for fast switching.