To turn off the narrator in Biomutant, you will need to visit the options section in the main menu. Once at the menu, click options and then navigate to the audio section - it will be right next to controls. Five options down and below the user interface option will be one for speech; turn it all the way down. This option controls the volume of the narrator, and setting it to zero will mute his voice and turn him off effectively in the game.
However, you will still see the narrator's subtitles show up during conversations, and it will also mute the voices of other characters in the game. So if you still want to hear character voices but just want the narrator to shut up for a bit, you will need to move down to the option called narrator under frequency.
This option controls the frequency in which the narrator will speak in the game. So use your controls to lower the tab, and doing so will reduce the number of times the narrator will speak in the game. Along with this, It will also allow him to still talk during cutscenes or scripted events.
For more on Biomutant, we recommend checking out both All Races in Biomutant: Best Breeds to Choose and All Mutations in Biomutant Explained: Best Mutations to Choose