What causes the opamp to become warm in this charger? [question updated]
If IC5A is being used as a comparator, there's your problem. Its output is driving a diode and a BJT's base-emitter junction with no provision for current limiting. If it ever attempts to drive its output high, the only limitation on current output will be its own current output capabilities, which are spec'ed as about 40 mA.

That 40 mA will generate heat within the op-amp, equal to 40 mA times the drop from its positive power supply terminal (not shown in the schematic) to about 1.4 V (two diode drops above ground).

Simple solution (for this problem, not necessarily all the problems in your circuit) is to replace the diode between the op-amp output and the BJT base with a resistor. The appropriate value is as high as possible, but low enough to ensure driving the BJT to saturation when the op-amp output is high. What value that is depends on design details you haven't shared.