Heat dissipation zener diode and resistor
Are you powering the circuit through the LM7805 when it's hooked up to the USB supply?  LM7805 requires about a ~2V headroom (ie +7.0VDC in in order to regulate +5.0 out at 1A).  So you may not actually be running that zener regulator at 5V input- check your 5V rail to verify.  With the battery you now have 5V and that resistor is going to get warm. Note that wattage ratings on resistors often assume a pretty high heat rise, I don't know what package (SMT/through hole) you're using but it can get uncomfortably warm with a 1/2W dissipation rating.