If a tenant of an apartment willingly accepts a roommate, can the roomate be served with an order of protection or arrested for tresspassing?
The wording of your question is a bit confusing and does not have enough facts to be able to definitively answer the question. What I can do is explain how the law works in most states.
First, let us look at the term "order of protection." An order of protection in most states is a court order that is used to remove a person who is a family member, a spouse, or a boyfriend or girlfriend, always a person who has abused or who has threatened to abuse. I do not know what state you reside in, but generally, a roommate is not a person who can be removed pursuant to an order of protection.
Second, usually, only a landlord may assert a claim of trespass because the premises are legally his or hers. However, in most states in which a tenant or someone living on the premises is there against the wishes of the landlord, the police will not arrest that person because the law requires an eviction action to remove a person living in a rental unit. If the police could act on a landlord's allegation of trespass, most landlords would be likely to bypass the usual mechanism for evicting people, which would deprive them of due process.
Because of the confusion in the terms you use, I cannot tell whether this is a situation in which someone has been abused or threatened by a roommate or whether this is a situation in which someone simply wants the roommate to be gone. If the roommate is boyfriend or girlfriend, an order of protection will result in that person's removal at least temporarily. However, if this is simply a situation in which there is a desire for the roommate to leave, the police will usually not act. If the roommate is present with the knowledge and consent of the landlord, it is up to the landlord to initiate proceedings to remove that person. If the landlord is unaware of the roommate's presence, the person who is the legal tenant is likely to be in violation of the lease, so seeking help from the landlord could very well result in the eviction of the tenant, too.
This sounds like a complicated situation, so it might be best to consult an attorney. Some cities have free legal services for those who qualify, sometimes a county bar association will offer discounted consultations, and universities with law schools often provide help through legal clinics. Your county bar association can help you find one of these resources, if they are available in your area.