I will work in the next month.
means that the day or days when you will work will occur some time within the next month. Perhaps you will work on November 4, or from November 8 to November 12, or just on November 3, 8, 14, and 29. Or perhaps you will start your next job on November 10 and continue working for the rest of November and beyond.
I will work for the next month.
means that you will work starting from approximately now or tomorrow, and the duration will be one month.
With no preposition:
I will work next month.
the meaning could be either "one or more times within November" or "the full duration of November". The context would need to indicate which meaning was intended, or it would be ambiguous.
Here are some similar contrasting examples with in and for:
My vacation lasts for a week. [The duration of my vacation is one week.]
My vacation begins in the middle of the week. [The starting day of my vacation is Wednesday or close to Wednesday.]
My vacation begins next week. [The starting day of my vacation is one of the days in the following week.]
I watched the groundhog every day for a year.
I only saw the groundhog twice in the past year.
I watched the groundhog last year. [Possibly as an ongoing activity lasting all year, possibly just once or twice during the year.]
The next Star Wars movie is scheduled to come out in 2019.
The next Star Wars movie is scheduled to come out in December, 2019.
I worked for George Lucas for six years. [The duration was six years.]
I worked for George Lucas in 1986. [My employment began and ended within 1986; possibly I had several terms of employment within 1986.]
I saw George Lucas last week. [I saw him one or more times during the week of October 6–12.]