Your concerns about living expenses are important and a necessary reality - however you are putting these concerns above the health and safety of your child. You are exposing your child to domestic violence - whether or not it's directed at the child or just in front of the child is irrelevant. It changes the child and stunts them. You will limit that child's possibilities and potentially make them grow up either a victim or an abuser.
I suggest that you get more information about domestic violence and educate yourself very well on this issue. If you continue to ignore big warning signs, you may one day find yourself losing your child completely. If you continue in the relationship, you may find yourself so beaten down emotionally, even physically, that you will not have the will to leave anymore.
Do not remove the protective order. Take care of your first and most important responsibility - be a protector for your child.
You and your boyfriend are in the best position to know if your relationship is healthy enough, for you and your child, to resume contact. There are ways to have a no contact order lifted. The easiest is simply to speak to the probation officer and make your case why he/she should lift it. If that fails, you can return to court and ask the judge to lift it. In either case, if working through the system becomes difficult, you can hire an attorney to help you lift the order. Good luck.