The 0.3V drop comes from the Schottky clamping diodes used to protect the pins of the chip. These diodes typically connect between each pin and the two power rails. If they are forward biased by more than 0.3V, arbitrarily large currents can flow.

The diodes are designed to absorb transient currents produced by ESD, which represent limited amounts of energy that they can handle, protecting the sensitive MOSFET gates from overvoltage. But if you drive them with a low-impedance source, you'll quickly dump more energy into them than they can handle.