Arduino Due's operating voltage is 3.3 volt and Ethernet shield operates on 5 volt supply. So my question is that this Ethernet shield will work on an Arduino Due and also if it will not harm the Arduino Due board due exceeding the operating voltage?
The Arduino Ethernet shield (or at least V2 of it) is based on a W5500 chip which itself features "3.3V operation with 5V I/O signal tolerance". In other words, the core of it already is a 3.3v device, just one that can tolerate connection to a 5v Arduino.
Of course the shield is more than just the chip. A very quick look isn't showing any glaringly obvious uses of 5v levels on the board in a place where a 3.3v Arduino derivative wouldn't expect them.
However, you should review the the full schematics and make your own determination that these are applicable to the specific board you plan to use and do not place 5v level on any signal the Due could not accommodate.
It's unfortunate that the shield documentation doesn't seem to say anything about compatibility, but worth noting the the Ethernet Library has some specifics for the SAM family of chips one of which is used on the Due.
The Ethernet Shield V1 was based on the W5100 chip. This may not have 5v tolerant inputs, as it appears to use level translation on the input side. But it is also powered from 3.3v, so likely the outputs would be fine for a Due. An interesting question is if the V1 shield would be damaged if connected to a 5v Arduino with a sketch that erroneously drove MISO high as an output.