What is the role of the pull-down resistor in a diode-transistor logic (DTL) gate?
R3 isn't strictly necessary as the transistor Q1 base will either be pulled up by R1 or pulled down by Da or Db, provided their Vf voltages are lower than the Vce threshold of Q1 (for example, if they're Schottky diodes.) If their thesholds are higher, R3 is needed to get Q1 to actually turn off.
R3 also helps with switching time: R3 and R1 can set a bias point for Q1 so that it isn't in full saturation when 'on'. Then when Da or Db pull low, the swing required at the base for Q1 to turn off is less.
Another way to ensure that Q1 turns off is to insert a diode in series with the Q1 base. Then Vce will be one diode drop less than Da and Db, ensuring that Q1 base current is at or near zero.
As follows (simulate it here):