Does coal production affect the production of other resources?

Yes, if you run out of charcoal in your warehouse (the total at the top of the screen) then the Efficiency of other rss production goes down. It will not stop altogether though. You should still be producing some charcoal from the Charcoal workshop.

You can top up the warehouse with any stored charcoal items they you may have acquired from rewards or from the trading float for a quick fix, but keeping the charcoal factories stocked with lumber will be the best option.