State laws and even individual judges handle child support differently. However, as a general rule, the parents are responsible for the finances required to raise their children. If you cohabitate with your boyfriend, it may help defray your cost of living and loosen your budget, but it doesn't change the responsibility of your children's father.
A child's two parents are responsible for his welfare. Nothing changes that throughout the raising of a child. Family court judges consider the finances of the parents when assigning child support amounts and may adjust amounts periodically based on changes in financial status, children's ages and personal circumstances. However, a boyfriend with separate finances who is not legally responsible for the children or even committed long-term cannot count as a rightful or reliable source of income and support.
If down the road you and your boyfriend become engaged and married, then a judge may reconsider the financial support assignments -- particularly if your children's father is struggling to meet his child support burden. Marriage typically brings a financial merger and extra means of support. If nothing else, it takes the strain off your income on a permanent basis. Therefore, your finances change and that's grounds for a child support adjustment.
A judge isn't likely to add financial burden on a mother if a child's father has a well-paying job and strong finances. Ideally, parents who are both active parts of their children's lives and take part in their upbringing share the financial burden. If a court feels a father has the means to contribute significantly to his children's welfare, chances are his obligation won't change because of your romantic life.
In the end, a parent can petition a family court for child support review for any reason at any time. Judges have tremendous discretion, particularly in family court where they are charged with sorting through the issues of human life and welfare as they balance application of the law with the needs of the children. Your children's father could learn that your boyfriend moved in and file a petition for a child support adjustment. A judge can decide whatever she thinks is appropriate.