To be fair, Eliot never actually uses the term "zombie" in this poem, but his descriptions of modern people going about their daily routines definitely feel zombie-ish. This might be because he usually describes these people by drawing from the Inferno and Purgatorio by Dante, two poems that describe the inner workings of hell and purgatory. You catch a glimpse of this type of crowd when the speaker observes in lines 61-63: "Under the brown fog of a winter dawn, / A crowd flowed over London Bridge, so many, / I had not thought death had undone so many." Like it or not, Eliot's basically slamming anyone who's content to go about his or her daily routine without thinking all that much about their lives. This sort of undead life represents the spiritual poverty of modern people, who "are now dying / With a little patience" (329-330).
Line 56: The fortune-teller Madame Sosostris tells you that she "see[s] crowds of people, walking round in a ring." This ring no doubt refers to the rings or "circles" of hell, which Dante describes as being filled with people who know they've got no hope of ever getting out. Can you guess whom Eliot is comparing these people to? You got it: pretty much everyone living in the modern world.
Lines 62-68: Here, Eliot really sinks his teeth into the zombie crowd image, describing the crowd of people that "flowed over London bridge, so many" (62). Again, he's referring to Dante's vision of hell. You can get a real sense of the despair Eliot sees in modern people, especially in the "Sighs, short and infrequent" that come out of them, who "fi[x] [their] eyes before [their] feet" (64-65). This idea of keeping your eyes on the ground in front of your feet is supposed to make you think about how you're always just worrying about the next thing to do. Modern folks never tend to lift their eyes to think about life (or even the world) as a whole. This is something Eliot would love for us to fix, but he's not all that optimistic about our chances.