What evidence is there to show conclusively that Hitler was or was not Jewish or had Jewish ancestry even if he was not a practicing Jew?
It's not known with full certainty who Hitler's father's father was (although George Hiedler is the generally accepted one). For a list of theories on the matter, see the Wikipedia article about Alois Hitler.
Since Hitler's father's mother got pregnant while allegedly working for a Jewish family it has been claimed that the father is actually the 19-year old son of this Jewish family. That would make Adolf Hitler one quarter Jewish according to the race laws in Nazi Germany.
However, there is not a smidgen of evidence for this theory. It has become popular because of the immense potential for irony, should it be true. But it probably isn't.
Source: Frank McDonough, Hitler and the rise of the Nazi Party, Pearson Education, 2003, p. 20.
Who is a Jew?
The definition of who is a Jew varies according to whether it is being considered by Jews based on normative religious statutes, self-identification, or by non-Jews for other reasons. Because Jewish identity can include characteristics of an ethnicity, a religion, and citizenship, the definition of who is a Jew has varied, depending on whether a religious, sociological, or ethnic aspect was being considered.
Genome-wise: He surely shared genome with the Jewish population. We all come from a common ancestor. Furthermore, there is no such thing as a Jewish race, so the question does not make much sense when seen from this point of view. Everybody is related to Jews.
Religion-wise: He had a Christian family and education:
Self-identification-wise: He didn't really like Jews, so I doubt he self-defined as one:
"If I am ever really in power, the destruction of the Jews will be my first and most important job. As soon as I have power, I shall have gallows after gallows erected, for example, in Munich on the Marienplatz-as many of them as traffic allows. Then the Jews will be hanged one after another, and they will stay hanging until they stink. They will stay hanging as long as hygienically possible. As soon as they are untied, then the next group will follow and that will continue until the last Jew in Munich is exterminated. Exactly the same procedure will be followed in other cities until Germany is cleansed of the last Jew!"
According to Nazi laws: unlikely (but the laws seem recursive to me, so it's really hard to say). I doubt that any German tribunal would have declared him a Jew in those times. For more information see Wikipedia.