Let's say it is January and I forgot what month is after it. Which one of the following sentences is correct?
What is next month?
What is the next month?
"What is the next month" is best, as if you were shortening to from "What is the next month after January?".
However, this is fairly informal.
Because you are asking the person to select one month from a group of options -as if they were selecting from a list of the months - a more formal and grammatically correct way to say it is, "Which month is next?"