GTA 5 nightclub production while playing heists
Yes, nightclub production continues so long as (a) you are playing the game in online mode (including heists) and (b) the stock for the particular good that you are producing is not full. You can purchase additional staff to produce more concurrent goods, and additional warehouse space to store larger inventory quantity. You can also upgrade the equipment to produce faster.
Some additional helpful information:
The equipment upgrade, though expensive, makes production go twice as fast. This applies across all of the staff you have hired.
You must have a corresponding MC business for each of the product types, plus the hangar for special cargo and the bunker for gunrunning.
A maximum of five staff can be assigned, but you can rotate them among the seven product types.
I've noticed glitches where one or two product types get stuck (Xbox One). Not sure what triggers this, and don't fully know how to reset, but I suspect rotating the staff off and back on will fix it (I'm letting it run now to see).
Goods tend to accrue fairly slowly.