Where? I would say that the probate court will recognize the name on the birth certificate as the father (whether he is or not).

Were ANY tests performed to prove who is (or is not) the father? Why is the non-biological father's name on the birth certificate? Was he married to the mother?

He was married to the mother. His name is on the birth certificate and neither he, nor the child knew he was not the biological father until the child was 35 years old.

Inherit from who, bio father or birth certificate ( legal father)

What location is it in ?

ANY question on law needs to include your general locale as laws vary by area.

However, in most areas, when a child is born to a married couple the husband is the legal father. While this can be challenged, 35 years after the fact is generally too late. Which means the father remains the legal father and the child would have rights of inheritance.

The initial determination of parentage is an essential factor in establishing heirship and probate succession.
The question of what factors should control in determining legal parentage raises critical concerns in probate law because the fact of parentage establishes a child's inheritance rights.One particular issue that has not been adequately addressed by many state legislatures is whether, and under what circumstances, a child with a presumed father and separate biological father may rebut the presumption of paternity in order to inherit from the biological father's estate./or to be disinherited by the fact of biology/ .

ANY question on law needs to include your general locale as laws vary by area.
So true.
The laws vary from state to state.
For example-New Jersey. There both UPA and UPC are adopted ,there is time limit for paternity proceeding , but New Jersey's Probate Code defers parental determinations to the Parentage Act, both statutes specifically provide that the twenty-three year statute of limitations to institute a paternity proceeding does not apply in probate cases.
Thus, under New Jersey's statutory scheme, paternity presumption may be rebutted at any time using genetic test .
/P.S.-That presumption may be rebutted by the child,mother or any interested party/

Wingate v. Estate of Ryan./NJ SuprCourt/ In Wingate,the court held that the Parentage Act's statute of limitations did not apply to a thirty-one year old claimant who sought to prove paternity and heirship under the Probate Code.