What is the purpose of the emitter resistor in this OR gate?
Either transistor can drive the output 'high', though this kind of rudimentary gate loses voltage from input to output, so a chain of many 'or' gates will, at some number, fail to function.

The resistor pulls the output low when both transistors are off.

Whether it will 'fail to work' without it depends on what it is connected to and how logic levels etc. are defined, so that's not an answerable question.