What happens to Jews who do not believe in Jesus?

Can a Jewish person go to Heaven if they don't believe in Jesus? Does being Jewish by itself automatically grant a Jew salvation?

Only by placing faith in the finished work of Jesus Christ may any human being receive God's forgiveness. This is true for Gentiles and Jews, so even a Jew must possess faith in Jesus to receive eternal life. As Paul says:

Rom. 3:28 For we maintain that a man is justified by faith apart from works of the Law.
Rom. 3:29 Or is God the God of Jews only? Is He not the God of Gentiles also? Yes, of Gentiles also,
Rom. 3:30 since indeed God who will justify the circumcised by faith and the uncircumcised through faith is one.

In the time before Christ's first coming, Jews were saved by faith in God's word that He would bring Israel a Messiah. Today a Jew is saved by faith in God's word that He did bring the Messiah, and the Messiah is Jesus. Apart from this faith, no Jew is saved (same as for Gentiles). Regrettably, many Jews do not have faith in Jesus as Messiah, which even Paul acknowledged in the first century:

Rom. 10:1 Brethren, my heart’s desire and my prayer to God for them is for their salvation.
Rom. 10:2 For I testify about them that they have a zeal for God, but not in accordance with knowledge.
Rom. 10:3 For not knowing about God’s righteousness and seeking to establish their own, they did not subject themselves to the righteousness of God.
Rom. 10:4 For Christ is the end of the law for righteousness to everyone who believes.

Despite the general lack of faith among the Jewish people, the Lord has promised to preserve a remnant of Israel in keeping with His promise to Abraham that he would have many (believing) descendants:

Rom. 11:1 I say then, God has not rejected His people, has He? May it never be! For I too am an Israelite, a descendant of Abraham, of the tribe of Benjamin.
Rom. 11:7 What then? What Israel is seeking, it has not obtained, but those who were chosen obtained it, and the rest were hardened;
Rom. 11:25 For I do not want you, brethren, to be uninformed of this mystery — so that you will not be wise in your own estimation — that a partial hardening has happened to Israel until the fullness of the Gentiles has come in;

God has always been at work preserving a remnant of believing Jews throughout the history of the nation. Paul explains that this was true even before Jesus came:

Rom. 11:2 God has not rejected His people whom He foreknew. Or do you not know what the Scripture says in the passage about Elijah, how he pleads with God against Israel?
Rom. 11:3 “Lord, THEY HAVE KILLED YOUR PROPHETS, THEY HAVE TORN DOWN YOUR ALTARS, AND I ALONE AM LEFT, AND THEY ARE SEEKING MY LIFE.”
Rom. 11:4 But what is the divine response to him? “I HAVE KEPT for Myself SEVEN THOUSAND MEN WHO HAVE NOT BOWED THE KNEE TO BAAL.”
Rom. 11:5 In the same way then, there has also come to be at the present time a remnant according to God’s gracious choice.

So while it is disheartening to realize how many Jews will not receive eternal life, nevertheless we are encouraged to see God's faithfulness in preserving His people despite their unfaithfulness to Him.