The answer is "It depends"
If the battery has a shorted cell or 2. YEs it might
If the battery is HIGH RESISTANCE no it won't so long as you are on shore power
If the battery is Low capacity.. It should not but the lights may go out swiftly if not on shore power.
IN short it all depends on the failure mode of the battery.
In all 3 cases the procedure is the same
Remove battery. Insulate positive lead and check converter (measure voltage) to insure operation.. IF NON_OP repalce converter and battery. If converter working normally replace battery. BUT DO CHECK THE CONVERTER.
OH. If Converter is a magnetek 6300.. UPGRADE to Progressive Dynamics 4600