Where Does VDD+0.3V Input Limit Come From on IC chips?
Most likely there is an ESD protection diode connected between the input pin and the VDD net on the chip, in such a way that it is normally reverse biased (A schematic showing the configuration is given in Peter Smith's answer). The idea is that when there is a positive ESD event, current will flow into the lower-impedance VDD net where it will do less damage than if it's all dumped on the one poor CMOS gate that's attached to the input pin.

Because the limit is VDD + 0.3 V it's likely in your device the diode is a Schottky type instead of a PN junction. With a PN junction, you'll usually see a limit of VDD + 0.6 V or so. 

If you were to apply an input voltage above VDD (by more than 0.3 or 0.4 V) to this device, you'd forward bias this diode, and draw a high current from your source. This might damage your source or, if the source can supply enough current, heat up the chip to the point of damage.

If you use a resistor to limit the current into the input pin under these conditions, you might find the circuit works fine. Or, particularly if the chip is a very low power one, you might find the whole chip (and maybe other things connected to the same VDD) are powered up through the input pin, which often leads to unintended behavior.