During gameplay, players typically take on the role of a specific character, and having that character constantly speak would break immersion and draw attention to the fact that the player is controlling a computer-generated character. Additionally, in multiplayer matches, players may be on different teams with different objectives, so having all of the characters talking would create a cacophony of voices and make it difficult to focus on the gameplay.
However, there are certain situations in the campaign missions where the main characters will have specific lines of dialogue that advance the story or provide tactical information to the player. These dialogue moments are usually triggered by specific events or actions in the game, such as encountering an enemy or reaching a certain objective. Similarly, in multiplayer matches, characters will often have short quips or call-outs when performing certain actions, such as killing an enemy or capturing a point.
So while the characters in Call of Duty do not continuously talk during gameplay, they still have their voices and use them in specific situations to enhance the overall experience without disrupting the player's immersion.