A good question. Some of the major themes of Renaissance literature are no longer actively engaged culturally. This is in part because the Renaissance succeeded. For example, many Renaissance writers fought to emphasize life on this world, rather than the emphasis on Heaven and the afterlife that pervaded medieval literature.
Other themes and works are actively engaged, but in different forms. For example, Dr. Faustus made a deal with the devil for power and knowledge. This sort of exchange shows up in many forms of literature today. In science fiction, fantasy, and detective fiction, people have to deal with dark forces for essential knowledge. Think of Tolkien's Lord of the Rings for example. The theme is re-fought in different ways, but it is still fought.
Other works and authors are influential still, and seem as new and profound as when they were published. Think of Machiavelli and the pragmatism of The Prince. He is sometimes seen as amoral, but the question of how political power can best be held is very much an active one today. Works like The New Machiavelli or Machiavelli on Modern Leadership show how relevant some period authors can be.
In a sense, the question itself reveals a somewhat problematic assumption, namely that everything you learn should somehow be "relevant" in the sense of comfortable and familiar. Realistically, if you only learn things that are "relevant" to you, you are actually not doing any real learning at all. Learning means precisely engaging with things previously unknown, and experiencing different cultures and periods in order to challenge your own assumptions and the limitations of your world view. It is precisely in being alien from your culture that works such as those of Shakespeare are valuable.
If you live in a culture that relies on technology for instant communication, Shakespeare helps you understand the art of conversation. The play of the rustic suggests a time when people entertained themselves, putting on plays, playing the piano, singing, or talking, rather than watching canned entertainment. All of Shakespeare's plays show very different family structures than twenty-first century western ones.
Perhaps the way to get the most out of Shakespeare is rather than demanding he be relevant to you (in a sense placing yourself at the center of the literary universe), instead asking which of the assumptions you and your friends share are challenged by reading Shakespeare; think about the way that encountering other cultures and perspectives enables you to examine your own preconceptions critically.
Maybe I wasn't very specific, but I meant what problems or themes from the Renaissance are still relevant today?
Hamlet, for example, is about inner psychology, which now is still relevant. A lot of teenagers feel a lot of pressure and stress, which can, in the worst cases, burn them out and break them down.
I was wondering if there were other themes or problems literary texts from the Renaissance took up that we still experience today, or whether most were problems that are not relevant anymore.