Firstly: 

There is nothing wrong with a Muslim opening his business on the day of the Muslim Eids (Eid al-Fitr and Eid al-Adha), subject to the condition that he does not sell anything that may help some people to disobey Allaah. 

Secondly: 

With regard to opening one’s business on days that the non-Muslims take as festivals, such as Christmas and Jewish, Buddhist or Hindu festivals, there is nothing wrong with that either, subject to the condition that you do not sell them anything that may help them in their sins, such as flags, banners, images, greetings cards, lanterns, flowers, coloured eggs and anything else they use in their festivals. 

Similarly he should not sell the Muslims anything that could help them to resemble the kuffaar on their festivals. 

The basic principle concerning that is that the Muslim is forbidden to commit acts of sin or help anyone else to do so. Allaah says (interpretation of the meaning): 

“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allaah. Verily, Allaah is Severe in punishment”

[al-Maa’idah 5:2]

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: “A Muslim should not sell things that will help the Muslims to imitate them (the kuffaar) on their festivals, such as food, clothing and so on, because that is helping in doing evil.” End quote from Iqtida’ al-Siraat al-Mustaqeem, 2/520 

And he said: ‘As for Muslims selling to [the kuffaar] on their festivals the things that help them to celebrate their festivals, such as food, clothing, herbs and so on, or giving those things to them, this is a kind of helping them to establish their haraam festivals.” 

It is narrated that Ibn Habeeb al-Maaliki said: “Do you not see that it is not permissible for the Muslims to sell to the Christians anything that has to do with their festivals, whether it be meat, condiments or clothing; they should not lend them mounts or help them in any part of their festivals, because that is like venerating their shirk and helping them in their kufr. The authorities should forbid the Muslims to do that. This is the view of Maalik and others, and I do not know of anyone who disagreed with that. 

Iqtida’ al-Siraat al-Mustaqeem, 2/526; al-Fataawa al-Kubra, 2/489; Ahkaam Ahl al-Dhimmah, 3/1250 

Shaykh al-Islam also said: If the things they buy are used to do haraam things, such as crosses, palm branches, baptismal fonts, incense, meat that has been slaughtered for anyone or anything other than Allaah, images and so on, then they are undoubtedly haraam, such as selling them juice for them to use as wine, or building churches for them. 

With regard to things that they use in their festivals such as food, drink and clothing, the basic principles of Ahmad and others suggest that these are makrooh, but does makrooh mean it is haraam, as in the madhhab of Maalik, or that it is discouraged? The most likely view is that it is makrooh in the sense of being haraam, for he does not allow the sale of bread, meat and herbs to evildoers who will drink wine with them, because these things are helping them to manifest the false religion and increase the numbers of people who will gather on their festival. This is worse than helping one particular person. Al-Iqtida’, 2/2/552 

Ibn Hajar al-Makki (may Allaah have mercy on him) was asked about selling musk to a kaafir, knowing that he is buying it to perfume his idols, or selling an animal to a kaafir knowing that he is going to kill it in an improper manner in order to eat it. 

He replied: 

It is haraam to sell in both cases, just as the scholars said:  If the seller knows that the buyer is going to use the goods for sinful purposes, it is haraam to sell them to him. Perfuming idols and killing animals to be eaten without being slaughtered properly are two major sins, even for them, because the more correct view concerning the kaafirs is that the rulings of sharee’ah are addressed to them just as they are addressed to Muslims. So it is not permissible to help them by selling them things that may help them to do these things. Similar to knowing in this case is thinking that that will most likely happen. And Allaah knows best. End quote from al-Fataawa al-Fiqhiyyah al-Kubra, 2/270 

To conclude: it is permissible for a Muslim to open his business of the days of kaafir festivals, subject to two conditions: 

1 – That he does not sell them anything that can be used for sin or that will help them to celebrate their festivals. 

2 – That he does not sell to the Muslims anything that will help them to imitate the kuffaar in these festivals. 

Undoubtedly there are specific goods that are used for these festivals, such as greetings cards, images, statues, crosses, and certain types of trees. It is not permissible to sell these things, or to bring them into the store at all. 

As for other things that may be used for this festival or for other purposes, then the business owner should do his best to work it out, and not sell them to people whose situation he knows or who thinks will most likely use them for haraam purposes or to celebrate that festival, such as clothes, perfume and foods. 

And Allaah knows best.