The voices, in addition to Burnham, are those of Pike, Spock, and Sarek:
Jeffrey Hunter as Capt. Pike from "The Cage"
One of the voices is Capt. Pike, played by Jeffrey Hunter, extracted directly from the TOS original pilot "The Cage":
PIKE: You bet I'm tired. You bet. I'm tired of being responsible for two hundred and three lives. I'm tired of deciding which mission is too risky and which isn't, and who's going on the landing party and who doesn't, and who lives and who dies.
Leonard Nimoy as Spock from "Space Seed" and "The Immunity Syndrome"
One of the sound clips of Spock that we hear is from TOS "Space Seed":
SPOCK: Insufficient facts always invite danger, Captain.
We also hear the following from "The Immunity Syndrome", in which Spock accuses McCoy:
SPOCK: You speak about the objective hardness of the Vulcan heart, yet how little room there seems to be in yours.
Sarek, as portrayed by James Frain in Discovery
We also hear James Frain's Sarek:
SAREK: I see an outsider. The child molded by wisdom and human compassion.
A script search confirms that this particular piece of dialogue does not appear in any Season 1 episode and so it is either discarded dialogue from Season 1, dialogue that may be included in a Season 2 episode, or something that was produced expressly for this teaser. A number of news sites agree that the voice belongs to Frain (e.g. TrekToday).