The first half of the fourth verse of "I Come and Stand at Every Door" exhibits anaphora (the repeated "I need no..."): "I need no fruit; I need no rice / I need no sweets nor even bread." I think its function here is simply to emphasize that the narrator "ask[s] for nothing for myself."
In the last verse ("All that I ask is that for peace / You fight today..."), there's a second vocal part (initially just doubling the lead vocal but then harmonizing with it). This extra voice highlights the earnestness of the narrator's request ("All that I ask...").