It depends on the jurisdiction. You haven't used a tag to indicate one, but the question gives the example of the US.
In general, there aren't any legal consequences for having a child out of wedlock.
what will happen for this person if he is on a non-immigrant visa?
There will be no change in the father's status. The child will be a US citizen (unless the father is a diplomat and the mother is not a US citizen who would meet the criteria for passing US citizenship to the child if the child had been born abroad). After the child's 21st birthday (assuming the child is a US citizen), the child will be able to submit an immigrant petition on the father's behalf, and if that petition is successful, the father can apply for an immigrant visa or for adjustment of status.
Is it a crime having a child from a person you are not married with, while you are married with another person?
Perhaps in some jurisdictions, but I am not aware of any. Certainly not in the United States.
Can this child get a birth certificate with no problems? Does the law accept this man and woman (boyfriend - girlfriend) as the child's legal parent?
It depends on the state. There is traditionally a presumption that married people only have children with each other, but that presumption can generally be overcome through various degrees of legal process, ranging from statements by the parents to biological paternity tests.
Does this child birth have any effect on the current marriage of man? I mean the court/law still verifies the current marriage?
It might motivate the man's wife to sue for divorce, and it might make the outcome of any separation or divorce less favorable for the man, but there is no automatic impact on the marriage, at least not in the US.
Could this child cause any kind of problem in the future for the father if/when he wants to change his status from non-immigrant to immigrant?
Not as far as I can see. The most likely impact would be if the child's existence exposes some lie that the father had made in some application for an immigration benefit. If the father doesn't lie to immigration officers, this can't happen.
Furthermore, to be clear, I'm thinking here of fairly specific and somewhat farfetched scenarios such as one in which the father falsely claimed on the visa application form not to have been to some country, and the mother is known to have been in that country at the time of the child's conception.