When you write in first person, a character is telling the story. So, yes, the narrative should be written in the voice of that character. (The first person narrator is often the main character, but not always.) A narrator can have a formal voice, as William Hundert does in "The Palace Thief," a short story by Ethan Canin:
My classroom was in fact a tribute to the lofty ideals of man, which I hoped would inspire my boys, and at the same time to the fleeting nature of human accomplishment, which I hoped would temper their ambition with humility. It was a dual tactic, with which Mr. Woodridge heartily agreed. Above the door frame hung a tablet, made as a term project by Henry L. Stimson when he was a boy here, that I hoped would teach my students of the irony that history bestows upon ambition.
A narrator's voice can be casual and colloquial, as Dulcie's voice is in "Crazy Life" by Lou Mathews:
Chuey is in this big cell, all by himself except for one other guy. When I see who that is, I know why Chuey's in trouble. Sleepy Chavez is sitting next to him. I don't know why they call him Sleepy. He's wired most of the time. I think he might have been a red freak once. Sleepy is one vato loco. The craziest I know. Everything bad that happens on 42nd Avenue starts with Sleepy Chavez.
Some writers use the distance of time to give themselves more latitude with their narrator's voice. ZZ Packer's short story "Brownies" tells of an event that happened while Laurel, the narrator, was a child at camp with her Brownie troop. The sophistication of Laurel's voice, however, makes it clear she's telling this story as an adult:
When you lived in the south suburbs of Atlanta, it was easy to forget about whites. Whites were like those baby pigeons: real and existing, but rarely seen or thought about. Everyone had been to Rich's to go clothes shopping; everyone had seen white girls and their mothers coo-cooing over dresses; everyone had gone to the downtown library and seen white businessmen swish by importantly, wrists flexed in front of them to check the time as though they would change from Clark Kent into Superman at any second.
Though Packer uses the more sophisticated voice, she doesn't include much of the insight Laurel must have gained about this childhood incident over the years. This keeps the attention on Laurel's youthful experience without limiting the voice to a child's language.
A distinctive voice is an integral part of first person; use it to create a rich and engaging narrative.