In Act 1, Scene 5, after hearing a great deal about how persistent Cesario is and how he looks too young to be a man and too old to still be a boy, Olivia becomes intrigued enough to allow him to speak with her, even though she is presently rejecting the company of any men but her kinsmen because she is deeply in mourning over the death of her brother. Since she is in mourning, she is dressed in mourning clothes, and one common mourning garment is a veil to hide any signs of weeping or sadness. Hence, Olivia finally grants Cesario permission to speak with her out of curiosity, but she firsts has Maria put a veil over her face, as we see in Olivia's line, "Give me my veil; come, throw it o'er my face. We'll once more hear Orsino's embassy" (I.v.155-56).
But in this case, the veil is serving two purposes for Olivia. The first purpose is to represent the fact that she is in mourning, but the second purpose is to actually conceal her face from any men. Since she want's nothing to do with any men out of mourning for her brother's loss, she also wants to hide her beauty from all men.