It might be my own myopia, but I don't think Shakespeare will lose relevance.
I think that Shakespeare will continue to have meaning on several different levels. One of them is thematic. The themes that emerge from Shakespeare's works are universal. The idea of lovers who cannot be together will forever be associated with Romeo and Juliet. Jealousy and insecurity are embedded parts of Othello, while the desire for power is an inseparable component of Macbeth. When parents struggle with their children, the words of King Lear resonate with clarity. Shakespeare will be thematically relevant because of his unique ability to comment on the human condition. People struggle with the realities he articulated. For this reason, I tend to think that people will always turn to him in order to better understand what it is they are experiencing.
Sometimes, turning to Shakespeare is forced upon us. We are exposed to Shakespeare in our schooling and it becomes one of the few elements that remain with us. I think that this is another reason why interest in Shakespeare will always be there. Formal education recognizes Shakespeare as a part of what students must learn when they are learning literature, drama, or even philosophical insight. Students at secondary, collegiate, and graduate level study Shakespeare with an intensity that convinces me that he will not be overlooked from a curricular point of view.