With regard to Hadd punishments, Jews and Christians are not punished with the Hadd for consuming Khamr (wine), though are liable for other Hadd punishments such as the Hadd for Zina (adultery), for the Hadd for Qadhf (false accusal of Zina) and for Qiyas (retaliation) because these have a basis in the Torah.
This is distinguished from the case of the Ahl al-Dhimmah, because
they have agreed to abide by our laws with respect to matters of
mu'amiallat like zina and riba. Further, the authority imposing
obligations is established due to the unity of the dar. According to
Abu Hanifah (God bless him), the Ahl al-Dhimmah have not undertaken
the obligation to follow our purely religious laws and those of the
muamalat about which their belief is different nor do they fall
under the authority established through the sword and domination. All
this is cut off for them on account of the contract of dhimmah. We
have been ordered to leave them alone as well as what they practice by
way of religion.37 They are, thus, like those in the enemy territory
except in the case of zina as it is prohibited in all religions. Riba
has been excluded from their contracts due to the saying of the
Prophet (God bless him and grant him peace), "Beware! Anyone who
indulges in riba, between him and between us there is no contract."
just as the Dhimmi has agreed to abide by them for his entire life.
Consequently, he is to be awarded the hadd of qadhf and is to be
subjected to qisas (retaliation), as distinguished from the offence of
drinking khamr, because its permissibly is part of his faith.
The basis of this are the following Quranic verses:
Quran 5:43 But how is it that they come to you for judgement while
they have the Torah, in which is the judgement of Allah ? Then they
turn away, [even] after that; but those are not [in fact] believers.
Quran 5:44 Indeed, We sent down the Torah, in which was guidance and
light. The prophets who submitted [to Allah ] judged by it for the
Jews, as did the rabbis and scholars by that with which they were
entrusted of the Scripture of Allah, and they were witnesses thereto.
So do not fear the people but fear Me, and do not exchange My verses
for a small price. And whoever does not judge by what Allah has
revealed - then it is those who are the disbelievers.
Quran 5:45 And We ordained for them therein a life for a life, an eye
for an eye, a nose for a nose, an ear for an ear, a tooth for a tooth,
and for wounds is legal retribution. But whoever gives [up his right
as] charity, it is an expiation for him. And whoever does not judge by
what Allah has revealed - then it is those who are the wrongdoers.
And the following Hadith:
From Saheeh Bukhari, also here and a more detailed version in Sahih Muslim:
A Jew and a Jewess were brought to Allah's Messenger (ﷺ) on a charge
of committing an illegal sexual intercourse.
The Prophet (ﷺ) asked them. "What is the legal punishment (for this
sin) in your Book (Torah)?"
They replied, "We only disgrace and flog them with stripes."
Abdullah bin Salam said, "O Allah's Messenger (ﷺ), tell them to bring
the Torah." The Torah was brought, and then one of the Jews put his
hand over the Divine Verse of the Rajam (stoning to death) and started
reading what preceded and what followed it.
On that, Ibn Salam said to the Jew, "Lift up your hand." Behold! The
Divine Verse of the Rajam was under his hand.
So Allah's Apostle ordered that the two (sinners) be stoned to death,
and so they were stoned. Ibn Umar added: So both of them were stoned
at the Balat and I saw the Jew sheltering the Jewess.
With the apparent meaning that the Jews would not have been stoned if the ruling wasn't present in the Torah.