Would airline got charge for late on depart from the gate due to passenger boarding issue? If so, what is the charge like?
Chain of delays

Being late on departure is likely to cause you to be late on arrival, this has the following immediate consequences:


you may be delayed even more on arrival in congested airports (e.g. London) if you miss your scheduled landing timeslot, being forced to circle around for, say, 20-30 minutes wasting fuel and time until you're allowed to land; I have been on a flight like that.
the main immediate effect is that some transferring passengers may be delayed because of this, which may possibly result in costs to you to reroute or compensate them.
you likely have scheduled the same airplane for the next flight with a minimal turnaround time; and delayed arrival may easily cause your next flight of that same airplane to be delayed (again, I have personally seen delays caused by this)
you likely have scheduled the same crew for the next flight, and in certain situations that delay may push them over the allowed work/rest schedule, so they are prohibited to fly their next (last in their shift) flight - so you need to rapidly find a new crew for the scheduled flight or cancel it.


For these reasons, a single delay sometimes might cause a chain of other delays or even cancellations, and become quite expensive.