The story lines of Shakespeare’s plays have been the basis for many works in the modern era—from plays to movies to TV shows. Think about the various storylines or plots in A Midsummer Night’s Dream and write about how you see a similar plot or plot device at work in a contemporary work that you have read or seen. For instance, can you think of another instance of star-crossed lovers and misplaced affections? Or, couples who plot to get the upper hand in a relationship, like Oberon and Titania do? Or, instances of comedy of manners like those we get with Bottom and his crew of mechanicals? (Please answer the question 300 words.)

What a great question! You are correct. Shakespeare's works have inspired a lot of remakes of his plays or retreads of story lines and characters that Shakespeare first created.
I'll start with the movie Get Over It starring Kirsten Dunst. The film is about a high school musical production of A Midsummer Night's Dream, and it comes complete with a love quadrangle motif. The film also includes some daydream sequences in magical forests which directly allude to the setting found in Shakespeare's original play. Nick Bottom's character is also reimagined in Martin Short's character. His character is a super untalented theater director with a huge ego about how great he is.
The question specifically gives an example of couples trying to get the upper hand in a relationship like Titania and Oberon. This concept brings to mind the film How to Lose a Guy in 10 Days. While the exact interactions between Andie and Ben do not exactly mirror Titania and Oberon, the two people are clearly trying to manipulate the other person's feelings.
A Midsummer Night's Dream brings up the concept of a love potion. Oberon wants Titania to be given a potion that will make her fall desperately in love with the first person that she sees. The concept of a love potion appears all over the place in modern-day media. In 1963, The Searchers released a song titled "Love Potion No. 9." The song is about a man that struggles with dating, and he is given a magical love potion to fix his problems.

I told her that I was a flop with chicsI've been this way since 1956She looked at my palm and she made a magic signShe said "What you need is love potion number nine"

The song and love potion concept was turned into a film by the same title of this song in 1992 with Sandra Bullock and Tate Donovan. The story revolves around two scientists that are hopeless when it comes to the opposite sex. They invent a substance that makes them irresistible to any person that hears them speak.
Finally, the short story "The Chaser" by John Collier includes a love potion as well. Alan Austen is the main character, who is frustrated that the girl of his dreams doesn't reciprocate his feelings. He consults a potion dealer, and the salesman sells him a potion that will make Diana fall madly in love with him.

"She will want to know all you do," said the old man. "All that has happened to you during the day. Every word of it. She will want to know what you are thinking about, why you smile suddenly, why your are looking sad."
"That is love!" cried Alan.