It depends. Some systems, where the external power brick is guaranteed to supply enough current to run the entire system AND charge the battery at max rate, disconnect the battery from the system when they connect it to the charger circuitry (eg: Rio Karma).

Other systems share input power between the system and the battery. There are 2 ways to do this:

1. All external power goes into the charger. System runs from battery only. If the system takes power during charge, it is in effect taking it direct from the charger (except when, say, there's a current transient so fast that the charger doesn't respond, in which case it comes from the battery). Obviously, charge time here is impacted by system load.

2. Have a priority setup where the system gets all its needs from the external supply, and any current "left over" between system needs and input current limit goes to the battery. Here, charge time *can* be impacted by system load, depending on what the max input current is and how much of this the battery charger is allocated. eg on an iPhone, the supplied brick is 5v/1A, and the battery charges at about 700mAish - so the system can take ~300mA without affecting the charge at all.

Neither of these should impact a full charge if the system is designed appropriately. If you are seeing what appears to be a lesser charge when the phone is not turned off during a charge cycle then either:

a) the charger is shutting off early and not retriggering (setup 1 above has issues of that sort, as the current drain never appears to stop and the charger's safety timeout can kick in, leaving the phone draining for a portion of the night)

or b) the battery metering system is getting confused and it really prefers a clean charge cycle with no drains in the middle.

Which phone?