From the Wikipedia article linked, the Pesharim approach posits that besides the surface meaning of an inspired text, there is a hidden or secret meaning that can only be exposed by individuals who have the requisite knowledge to uncover that meaning. Often the technique repurposes prophesy to apply not to the historical setting in which it was written, but to a contemporary setting.
A striking example is the Pesher to Habakkuk, which substitutes Roman oppressors for Babylonian or Chaldean oppressors in Habakkuk. The Pesher (or commentary) is interlaced with the original text so we can clearly see that the surface meaning remains a prophesy about the Chaldeans. But the secret interpretation refers to the Romans. The two nationalities are apparently spelled similarly in Hebrew, which seems to be the justification for making the substitution.
The specific interpretations of the Qumran community probably have no modern legacy for either Christians or Jews. The texts themselves were lost to history until very recently as nobody found them valuable enough to copy once the Qumran people disappeared. If the Dead Sea Scrolls are the product of the Essene sect of Judaism, as many suspect, their legacy died out around the time of the destruction of the temple in 70. If they aren't Essene, we can't trace their legacy at all.
However, the general concept of secret meanings in inspired texts is alive and well in modern Christianity and in the Jewish Kabbalah tradition. Gnosticism is the general umbrella term that covers these sorts of beliefs. One recent example of hidden knowledge is the idea popularized by The Bible Code that the Bible contains encrypted messages. As with many other examples, the primary focus seems to be on prophecy.
The idea isn't entirely foreign to Paul who writes in Ephesians 3:4–6 (NASB):
By referring to this, when you read you can understand my insight into the mystery of Christ, which in other generations was not made known to the sons of men, as it has now been revealed to His holy apostles and prophets in the Spirit; to be specific, that the Gentiles are fellow heirs and fellow members of the body, and fellow partakers of the promise in Christ Jesus through the gospel,
So Paul finds hints in the Hebrew Bible that point to Jesus as a messiah, not just for Israel, but for Gentiles as well. According to this text, the mystery has been laid bare to the Church by a revelation from the Holy Spirit. Perhaps texts like this one led to the rise of Christian Gnosticism which found mysteries in all sorts of texts. Paul seems to address the proliferation of belief in secret knowledge in Colossians 2:1–4 (NASB):
For I want you to know how great a struggle I have on your behalf and for those who are at Laodicea, and for all those who have not personally seen my face, that their hearts may be encouraged, having been knit together in love, and attaining to all the wealth that comes from the full assurance of understanding, resulting in a true knowledge of God’s mystery, that is, Christ Himself, in whom are hidden all the treasures of wisdom and knowledge. I say this so that no one will delude you with persuasive argument.
In other words, he says that all the mysteries of the Hebrew Bible point to Jesus and later in the chapter dismisses practices that arise from finding or knowing about other secret knowledge.
Modern Bible scholarship tends to rely on the historical-grammatical method, which strives to find the meaning originally intended by the author and therefore rejects Gnostic interpretations as a general rule. The most usual exception is the one suggested by Paul himself: that hints and precursors of Jesus may be found in the Old Testament. (Obviously most Jewish scholars reject this exception.)