Yes, there could be a feasible mechanism for this, although it isn't necessarily for heat dissipation for whole lasing units (which could just be static and fired in alternation), but based around some pseudo-science of how the imagined laser mechanism works.
Two important parts of a laser mechanism:
A primary source of incoherent energy (often light energy). For powerful military lasers, these sources can be unfamiliar such as a chemical laser - i.e. a laser that uses up a limited supply of ammunition. The primary sources can be high power, short-duration flashes resulting in pulse lasers - and this is a common restriction at higher power ratings.
A lasing cavity. A unit which absorbs the primary source and re-emits it coherently.
If you set up your imagined weapon with the following restrictions:
- The lasing cavity component is very expensive, maybe also heavy and/or bulky, but robust and can operate more or less continuously in theory, provided the energy keeps coming in.
- The primary energy source needs to go through a charge/release cycle, and charging takes significant time.
- The primary energy source may also need to cool after discharging to keep the mechanism within operating temperature.
- The primary energy source needs to be closely coupled to the lasing cavity when firing.
- The primary energy source needs to be closely coupled to a charging system to charge.
None of the above are unrealistic from a scientific point-of-view. It depends on the power requirements of the weapon - which is likely to be high, therefore options become restrictive and if you go beyond known technology you can invent/decide which apply within a science fiction world.
If you apply the above design restrictions, it may be worth having a central cavity surrounded by multiple primary sources that are used in turn. That doesn't guarantee that it makes sense to also physically rotate them - for projectile weapons this arrangement is often so that projectiles can share a firing mechanism and/or barrel, which has several benefits (such as saving weight)
However, this could lead to some mechanics which physically move the primary energy sources and for example, vent heat/air around a central barrel in rotation to assist cooling whilst a source is being charged, then move it back into place to couple with the lasing cavity. The advantage of such a set up would be more rapid firing, at lower cost than replicating the whole laser mechanism. In terms of physical appearance and sci-fi cool factor, the effect could be much the same as a rotating unit.
You could place one further restriction that would make a rotating system more viable:
- The nature of the lasing cavity means that only one coupling point is practical between the energy source and itself. That would mean if you want to increase rate of fire, you would need to have each energy source charged up then moved into position in turn. Variations of rotating sources, belt-feed or other mechanisms could then look good for getting charged energy sources available quickly.