It is important to realize that not everyone follows the same schedule or holidays and, if you do business internationally, you may have clients that fully expect you to ship out to them on days that may be considered holidays here in the US. On the other hand, many companies now operate every day of the year. As such, you may have to ship customer orders regardless of what day it is. The trick is finding a courier company that will be willing to pick up and ship on holidays. There are some couriers who are open every day of the year, like most medical courier companies, but there are also many good business couriers that are now open year round as well.
Usually, a courier company that is open every day of the year is going to use this element of their business as a strong part of their marketing strategy. That means that almost everything that has their name on it is likely going to have something like "open 365 24/7!" all over it. Usually, the website of a company is going to be the best place to get information about what days and hours they are open.
There is also usually a large contingent of couriers that while they may not be fully operational during every day of the year will at least have the option for emergency deliveries to be made. They may have a 24 hour emergency request line that you can call if you have a delivery which absolutely can't wait. They will have a few drivers on call at all times to respond to such requests, even though the whole company may not be open.
If you're making a request for a holiday pickup from a company that doesn't specifically advertise themselves as offering holiday service, be aware that you're very likely going to have to pay a surcharge in order to have the pickup made on a holiday. The courier company is going to have to pay their drivers extra, and they will make up that extra cost by charging it to you.
Keep in mind, that even though a holiday pickup might cost a little extra, it can really impress clients if you can respond to their emergency situations even during times when most other businesses are closed. Sometimes the extra money you spend on a courier for a holiday pickup is worth what you gain in customer satisfaction.