Preface: I'm getting most of this information from Flint and Vanderkam's The Meaning of the Dead Sea Scrolls and a bit from my own research.
For background on the Dead Sea Scrolls, see my answer here: http://goedhartvoordieren.nl/?page=r/AskHistorians/comments/5sz5mi/what_do_we_know_because_of_the_dead_sea_scrolls/ddj8ife/
And for background on the question of whether the DSS were written by the Essenes, see here: http://goedhartvoordieren.nl/?page=r/AcademicBiblical/comments/3z27v7/identification_of_qumranites_as_essenes_yay_or_nay/cyiwby8/
Now, I'm actually in the middle of a paper on an apparent controversy between most of these groups (Pharisees/Rabbis, Essenes, Christians, Hasmoneans) in the centuries surrounding Jesus, so I don't have as much time to spend on this as I'd like to, but I'm hoping I can offer some insight nonetheless.
This was a big question after the initial dissemination of some of the Dead Sea Scrolls, and in fact goes back at least as far as 1790, when a writer by the name of Karl Bahrdt "tried to account for the mysteries in Jesus' life by suggesting that he was a "secret agent" of the Essenes" (F/K 321). Ernest Renan had a similar take, that Jesus was trained by the Essenes, in 1863.
When the Scrolls first came to light, there was something of an explosion of scholarship about Jesus and Qumran. Most people didn't know many details about them other than the facts that there were biblical scrolls and peculiar sectarian scrolls, and that they ranged from the third century BCE to the first century CE. Not only that, but there was this mysterious "Teacher of Righteousness" that was referenced in a lot of the scrolls. Some responses were measured and some were outlandish. In 1950, André Dupont-Sommer wrote a measured analysis:
Everything in the Jewish New Covenant heralds and prepares the way for the Christian New Covenant. The Galilean Master, as He is presented in the writings of the New Testament, appears in many respects as an astonishing reincarnation of the Teacher of Righteousness. Like the latter He preached penitence, poverty, humility, love of one's neighbour, chastity. Like him, He prescribed the observance of the Law of Moses, the whole Law, but the Law finished and perfected, thanks to his own revelations. Like him He was the Elect and Messiah of God, the Messiah redeemer of the world [...]
That being said, he writes in the work that the Teacher of Righteousness, who most likely died nearly a century before Jesus, is useful for contextualization of the Jesus movement more than any kind of real spiritual forerunner.
Then things go off the rails a bit with the speculation. John Marco Allegro, who was on the original team of Dead Sea Scrolls editors, is a curious case of an academic mind going haywire. He started out quite measured, writing in 1956,
We should be wise at this stage to avoid too dogmatic assertions about the life of the Teacher or the manner of his death, or make too sweeping comparisons or contrasts with the Christian Master.
But then, as his views became increasingly odd, he came out in 1970 with The Sacred Mushroom and the Cross: A Study of the Nature and Origins of Christianity Within the Fertility Cults of the Ancient Near East, in which he "proposed that early Christianity was a orgiastic fertility cult that made use of a hallucinogenic mushroom containing the drug psilocybin. Moreover, Jesus never actually existed, but was invented by early Christians under the influence of this drug."
Another author, Barbara Thiering, wrote that all characters referenced in the DSS (The Wicked Priest, Teacher of Righteousness, etc.) corresponded to characters in the early Christian movement. She claims, according to F/K, that "Jesus marries Mary Magdalene twice, he is crucified near the Qumran complex but does not die, his unconscious body is placed in Cave 8, and he later lives out his days in Rome as an old man" (326).
I'm getting off track talking about the most extreme speculations (mostly because I find them fun to read about).
Many scholars have noted the similarities between John the Baptist's lifestyle and teachings and those of the Qumranites. Flint and Vanderkam note four things specifically:
1) John's family background and lineage: John was born to elderly parents. According to Josephus, Qumranites didn't marry, but they would adopt children, regard them as kin, and bring them up in their teachings.
2) Location of John's ministry: John taught in the Judean wilderness and it would be fair (though not sure) to assume that he came in contact with the sect at Qumran.
3) John's ministry shared many features with the Qumranites: An urgent message that the time was at hand, and the prominent place of baptism.
4) John's interpretive methods: they use similar passages of scripture as proof texts of their own teachings.
Now this is where scholars (myself included) would put forth something of a fork in the road. There is evidence (which I will get to presently) of similarities between aspects of the gospels and the writings at Qumran, but does this mean (A) that Jesus was an Essene or had studied with the Essenes or (B) that one or more of the gospel-writing communities had roots in the Essene movement? I suppose it depends on how faithful you believe the gospels are to Jesus' actual person and teachings. Anywho, here's a list of DSS passages that sound like the gospels:
1) In the Messianic Apocalypse (4Q521), we read, "For He shall heal the critically wounded, He shall raise the dead, He shall bring good news to the poor, He shall lead the holy ones, and the hungry He shall enrich [...]
2) In 4Q246 (Apocryphon of Daniel), we read, "Also his son will be called great, and be designated by his name. He will be called the Son of God, they will call him the son of the Most High [...]"
3) 4Q525 (Blessings of the Wise or Beatitudes): "Blessed is the one who speaks truth with a pure heart and does not slander with his tongue. Blessed are those who hold fast to its statutes and do not hold fast to the ways of injustice. Blessed are those who rejoice in it, and do not burst forth on paths of folly. Nlessed are those who seek it with pure hands and do not search for it with deceitful heart."
I need to cut it short here because I've lost some valuable writing time, but I'd suggest you pick up the Flint and Vanderkam book (A) because it's great and (B) because it has a good chapter on this question. I'll close out by suggesting that you read some of the Dead Sea Scrolls (including the Thanksgiving Hymns), and then read the Gospel of John. It's fascinating to read in sequence, and you can find lengthy conference proceedings on the subject.
In conclusion, there are arguments that could be made for Jesus having learned something from the Essenes, or at least from John the Baptist who might have learned it from the Essenes, but it depends how much you're willing to speculate. For me, the likeliest possibility is that, supposing some or all of the gospel-writing communities were originally Jewish and decided to follow Jesus's interpretation of the Law instead, it stands to reason that some of those Jews would have been of the Essene persuasion. I'm personally of the mind that the community that put forth the Gospel of John was more than likely coming from this background, based on the similarities in their methods of interpreting Hebrew scripture. But that's a discussion for another time. I hope this wasn't insanely long and boring!