Can power supplies cause temperature fluctuations...

Your question is too non-specific for a good answer. In general yes to all those questions. But whether these sensitivities and fluctuations are big enough to make a difference in the case you are thinking about, we can't say.

If you add more detail to your question, we may be able to give a better answer.

 

Zener diodes are sensitive to temperature change. So for example, if a zener were used as a regulator, or as a reference for a regulator, it will influence the voltage regulation of that specific rail.

Resistors and capacitors are also sensitive to temperature change, but this is much more attributed to their properties (as indicated on manufacturer spec sheet). Like a resistor has a temperature operating range and will have different resistance according to its properties.

The regulation issues will generally show up as voltage related and not so much show itself as an effect on current draw.

Another aspect is the proximity of heat generating components from the power supply to the circuit. If the circuit is co-located with the power supply, there will be a heat interaction between the two. How much that affects the circuit will depend on how close heat/EM generating are to other more sensitive components.