I don't think they directly affect the colours.
Indirectly, aperture and exposure can affect ISO. Colour saturation decreases as ISO increases.
Long exposure also heat up the sensor creating more noise, this will also have an impact on the colours.
Edited to add:
When photographing badminton games, the colours of each photo will appear to be different. This is due to the flickering of fluorescent light that is used to light the indoor arena. As far as I know, tungsten light does not flicker.
This create an interesting situation where the colour temperature will change for every shot taken when the shutter speed is modestly high.
Fluorescent light flicker at a rate of 60 hertz (as per the current) so in theory a shutter speed of less than 1/60 of a second is subject to sudden change in colour temperature (blue/green in one shot and yellow in another).
The cause is the flickering in the light and its obviously not the shutter's fault.
It may be a stretch to call this highly relevant but this is the only situation I can think of where shutter speed is linked with colour temperature.
Using a long exposure time may make it less of a problem but that is hardly an option for sports photography and especially badminton where shuttles travel at over 300 km/h.
However it is nice to keep in mind when photographing flickering light source that your shutter, if set to a fast enough value, can catch a gap between flickers, which is not noticed by the naked eye.