Does the prostration of forgetfulness (sujood as-sahw) become void for the worshiper if he forgets a transitional takbeer, and must the worshiper repeat the prayer in this case? What are the cases in which the prostration of forgetfulness is obligatory and when it is not?

What you did was not correct, but you are not required to repeat it as long as you did it out of ignorance. This is if the forgetfulness occurred while you were praying behind the Imam, which is apparent from the question. However, if you were a latecomer to the prayer and the forgetfulness occurred after you had separated from the Imam, then you were correct in performing the prostration of forgetfulness (sujood al-sahw).