DMX on the Arduino without additional driver chip
The physical layer of a DMX interface is based on differential signaling over an RS485 link.  This is exactly what is said here on the DMX Arduino.cc page.

A differential signal (like RS485 or USB) can travel greater distances than signals which reference ground (like RS232, I2C or SPI).  This is because noise, which affect both sides of a differential signal, can be canceled out at the far end when one of the signals is inverted then added to the other by the differential receiver.

Connecting one side of a differential receiver to ground and driving the other side with logic level voltage is not recommended.  Even if it worked and did not cause harm to the transmitting processor nor the differential receiver the signal will likely be susceptible to undesirable noise.

RS485 is a ubiquitous standard and likely several easily sourced designs exist for the Arduino.