One specific effect of using a first person narrator comes in the natural limitations this narrative form creates.
Just as Nick's access to the truth about Gatsby is limited, the reader's information is also limited. This dynamic allows the writer to carefully control the flow of information, creating anticipation and suspense.
In the case of this novel, Nick's limited perspective allows the mystery surrounding Gatsby to remain intact.
We see how the control of information can impact the narrative in several places in the novel, regarding Gatsby and, critically, at the novel's climax.
The details and circumstances of Myrtle's death are carefully doled out and details are withheld for maximum narrative effect. An omniscient narrative voice would not easily allow for the same style of narrative effect.