Represents your internal conflicts.
Reflects your lingering worries for what may happen to other people.
On one hand, if this is a descriptive dream, it means that you are having anxiety or fears of someone you know can eventually be hurt in real life.
On the other hand, a known person in your dream can represent an aspect or characteristic of your personality that is being affected in some way. Try to think of an attribute that characterizes that person in real life, for example, if the snake bites a girl, she can embodies your tender and fragile side. If the snake bites a friend of yours, it means that you may have some issues with your empathy or ability to relate to your friends, since your pal in your dream may be personifying the idea or friendship or social relationships. If the snake bites a random guy in the office, it could express problems or fears related to your work environment or your job duties.
Another interpretation says that the snake hurting a random person in the dream would represent a personal defect, an emotional weakness, maybe a bad habit in the dreamer that becomes uncomfortable, hurtful or offensive to your partner or dear ones. Certainly, this dream maybe a manifestation of introspection done by the dreamer’s unconscious mind. Having said that, I invite you to ask yourself if this interpretation is the most suitable, only you know it.