Considering as a whole how Jesus lived his life and interacted with all people, from his family and neighbours to the most despised people in his community, it seems unlikely that he wanted his disciples to live the life of a hermit, severing all their connections to humanity. This is not following his example at all.
But it is clear that his words, on more than one occasion, specifically aim to destroy what is considered our most significant connection to the rest of humanity - the genetic bond between family members. Why would he instruct us this way? More importantly, is it even possible to hate someone because they share your genetic code? How do we even begin to do that?
What does it mean to hate?
What is the nature of hate? What does it mean?
To hate is to destroy the relationship from our end, to intentionally disregard the feelings of the other person and directly or neglectfully cause them pain and anguish.
Did Jesus ever hate?
The automatic answer to this is no. But there are two passages that bear considering from the other person's point of view in order to understand their relevance to this question.
The first one is from early in Jesus' life, as told in Luke:
41 Every year Jesus’ parents went to Jerusalem for the Festival of the Passover. 42 When he was twelve years old, they went up to the festival, according to the custom. 43 After the festival was over, while his parents were returning home, the boy Jesus stayed behind in Jerusalem, but they were unaware of it. 44 Thinking he was in their company, they traveled on for a day. Then they began looking for him among their relatives and friends. 45 When they did not find him, they went back to Jerusalem to look for him. 46 After three days they found him in the temple courts, sitting among the teachers, listening to them and asking them questions. 47 Everyone who heard him was amazed at his understanding and his answers.48 When his parents saw him, they were astonished. His mother said to him, “Son, why have you treated us like this? Your father and I have been anxiously searching for you.”
49 “Why were you searching for me?” he asked. “Didn’t you know I had
to be in my Father’s house?” 50 But they did not understand what he
was saying to them. (NIV Luke 2: 41-50)
When Jesus stayed behind without telling his parents, knowing that he was expected to travel home with them after the festival, he disregarded their feelings as well as the expectations they had in terms of their relationship with him as his parents. From his parents' response, it's clear that they felt mistreated, and that he had caused them anguish. His reference to his 'Father's house' in reply would have also been painful for Joseph to hear - a cutting remark that completely disregards his own position as the boy's father.
This second passage from Matthew gives us another example of how Jesus treated family members:
While Jesus was still talking to the crowd, his mother and brothers
stood outside, wanting to speak to him. Someone told him, “Your mother
and brothers are standing outside, wanting to speak to you.”
He replied to him, “Who is my mother, and who are my brothers?”
Pointing to his disciples, he said, “Here are my mother and my
brothers. For whoever does the will of my Father in heaven is my
brother and sister and mother.” (NIV Matthew 12:46-50)
Here Jesus is very clearly downplaying the significance of his genetic connection to the people standing outside. Can you imagine how it would feel to be his brother and hear that response? To hear your mother and yourself being dismissed so readily by her eldest son, your brother, in front of crowds of people who would be guided by his example?
Both of these passages describe behaviour from Jesus that can be interpreted on the receiving end as hating his biological mother, father and brothers. Although he expresses no hate for them as human beings, he deliberately dismisses the significance of his relationship to them as a blood relative, as their son, and relegates them to the same significance of every other human being in his life.
What if you were told tomorrow that your parents are not your biological parents - that you have no actual genetic bond with the family in which you grew up? Putting aside the significance of a sense of historical identity, would that news change your love for them even one iota? Is there really such a thing as an emotional or spiritual connection built into our DNA, or is it simply our perception that intensifies the bond? Is it possible to lovingly connect with another person just as much, whether or not they share your genetic code? Can a mother love a child that is not her own just as much as if he was?
Evolutionary instincts to survive, to procreate and to benefit our own kind can generate physiological responses that intensify the awareness of a spiritual/emotional connection to another human being. What Jesus teaches by his example and by his blatant disregard for the genetic bond (and sexual attraction, for that matter) is that our spiritual potential to love another human being is actually the same for every human being, regardless of how close/similar/attractive they are to us genetically or physically, let alone culturally or ideologically.
Love as a spiritual connection is everywhere - only our awareness of it differs.