For example: 私には夢がある (I have a dream) Does using に have a different implication from ommiting it? Is it even possible to omit it in this sentence?
In sentences using ある/いる, に is usually used as a describer of location. Therefore I think a more precise translation would be "I have dream in me" or "There is a dream in me" (even if those sound a little weird in English).
I think the inclusion of に emphasizes the metaphorical holding of a dream within oneself, as opposed to just simply saying a dream exists. I would think that に could be omitted from this sentence as 私は～がある is a pretty common structure to describe things you have such as a car.