First I must correct you Zakat can also be given to non-Muslims and they are even among the fourth people in the ranking of the verse on Zakat (note as this is not part of the question I will stop here for details you may seek on the site or ask a new question or read here in qtafsir):
Zakah expenditures are only for the poor and for the needy and for those employed to collect [zakah] and for bringing hearts together [for Islam] and for freeing captives [or slaves] and for those in debt and for the cause of Allah and for the [stranded] traveler - an obligation [imposed] by Allah . And Allah is Knowing and Wise. (9:60)
Secondly as we know from the Qur'an that:
Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them. Indeed, Allah loves those who act justly. (60:8)
and from the sunnah:
Gabriel impressed upon me (the kind treatment) towards the neighbour (so much) that I thought as if he would soon confer upon him the (right) of inheritance. (Sahih al-Bukhari and Sahih Muslim and other hadith compilations)
So given parts of a qurban or feeding it to a neighbour or needy person (no matter which religion he has) or a non-Muslim who is not clearly fighting Muslims is considered as permissible. See for example islamqa #180503.
Nevertheless scholar hold different opinions:
The malikis considered it as makrooh and Imam Malik said "Others are more preferable to us".
The shafi'is consider it as haram for a qurban which is wajib, and makrooh for one that is mandub.
In this fatwa from islamweb #29843 after quoting the above views and adding that some scholars consider it as permissible to do it in case of non-Fighting disbelivers and neighbours, the mufti came to the conclusion that it is not permissible as nowadays many Muslims are in need, saying how can a Muslim feed a disbeliever when his brother is in need?
Ibn Qudamah -hanbali scholar- quoted in his al-Mughni -see here in Arabic-:
In the following I will translate from Arabic, as this is my own translation take it carefully
ويجوز أن يطعم منها كافرا وبهذا قال الحسن وأبو ثور, وأصحاب الرأي وقال مالك: غيرهم أحب إلينا وكره مالك والليث إعطاء النصرانى جلد الأضحية.
It is permissible to feed a disbeliever from it; in this is the view of al-Hassan, abu Thawr and the people of opinion (ashab ar-Ra'y). And Malik said "others are more preferable to us" and he and al-Laith considered it as frowned upon to give the leather of an odhiyah to a Christian.
ولنا أنه طعام له أكله فجاز إطعامه للذمى كسائر طعامه ولأنه صدقة تطوع, فجاز إطعامها الذمى والأسير كسائر صدقة التطوع فأما الصدقة الواجبة منها فلا يجزئ دفعها إلى كافر لأنها صدقة واجبة, فأشبهت الزكاة وكفارة اليمين.
While we consider it as a food for him wich he might consume, so it is permissible to feed a Dhimmi as it is the case for any of his food, because it is a volunteer sadaqah, so it is allowed to feed a Dhimmi or a prisoner of war as it is the case for any volunteer sadaqah, but in case of a sadaqah which is ordered it is not valid to give it a disbeliever as it is an ordered sadaqa and has to be considered similar to a-Zakat or kaffarah of an oath.
So if you had to offer for example as hajj who needs to offer hady it isn't permissible to give from the qurban or feed it a non-Muslim. According the view of the hanbalis who also seem to reject Zakat for non-Muslims in general.