Okay, here is where my theory of fatherhood is coming from:

You are not your genes. Your child is not your genes. Before people knew about genes, men knew that it was very important for them to get their semen into women, and that the resulting children were special. If a man's semen didn't work, or if his wife was impregnated by someone else's semen, the man would be humiliated. These are the values of an alien god, and we're allowed to reject them.

Consider a more humanistic conception of personal identity: Your child is an individual, not a possession, and not merely a product of the circumstances of their conception. If you find out they came from an adulterous affair, that doesn't change the fact that they are an individual who has a special personal relationship with you.

Consider a more transhumanistic conception of personal identity: Your child is a mind whose qualities are influenced by genetics in a way that is not well-understood, but whose informational content is much more than their genome. Creating this child involved semen at some point, because that's the only way of having children available to you right now. If it turns out that the mother covertly used someone else's semen, that revelation has no effect on the child's identity.

These are not moral arguments. I'm describing a worldview that will still make sense when parents start giving their children genes they themselves do not have, when mothers can elect to have children without the inconvenience of being pregnant, when children are not biological creatures at all. Filial love should flourish in this world.

Now for the moral arguments: It is not good to bring new life into this world if it is going to be miserable. Therefore one shouldn't have a child unless one is willing and able to care for it. This is a moral anti-realist account of what is commonly thought of as a (legitimate) father's "responsibility" for his child.

It is also not good to cause an existing person to become miserable. If a child recognizes you as their father, and you renounce the child, that child will become miserable. On the other hand, caring for the child might make you miserable. But in most cases, it seems to me that being disowned by the man you call "father" is worse than raising a child for 13 or 18 years. Therefore, if you have a child who recognizes you as their father, you should continue to play the role of father, even if you learn something surprising about where they came from.

Now if you fiddle with the parameters enough, you'll break the consequentialist argument: If the child is a week old when you learn they're not related to you, it's probably not too late to break the filial bond and disown them. If you decide that you're not capable of being an adequate father for whatever reason, it's probably in the child's best interest for you to give it away. And so on.