Donkeys have been "over-exploited" and their numbers needed to be kept at a sustainable level, the government said.
China is a big importer of donkey skins from Burkina Faso, using them to make traditional medicines.
In Burkina Faso, donkeys are commonly used to transport goods, though some communities eat their meat.
The surge in exports has put pressure on the local market, with the price of donkeys rising, says BBC Afrique's Abdourahmane Dia.
The price of the animal's hide has risen from about $4 (Â£3) a few years ago to $50, he adds.
Gelatin made from donkey skin is highly prized in China as a medicinal tonic, thought to nourish the blood, boost the immune system and act as a general pick-me-up.
It is sometimes referred to as one of the "three nourishing treasures" (zi bu san bao), along with ginseng and the antlers of young deer.
The most famous donkey gelatin is produced in Dong'e County in north-eastern Shandong Province, where it is traditionally made with the local well-water.
Donkey gelatin is sometimes mixed with walnuts, goji berries and other tonic foods and sold in dark, gummy slabs that can be eaten as a snack.
Health and longevity is a Chinese national obsession, and tonic foods like this are often lavishly packaged and presented as expensive gifts.
Aside from gelatin, donkey meat is a delicacy in some Chinese regions, especially in the north of the country.
Here, the lean meat is often simmered in a richly-spiced broth, and then cooled, sliced and served with a refreshing dip of chopped garlic and vinegar.
The meat has a fine, dense texture and a marvellous flavour that's a little reminiscent of beef.
In northern Hebei province, one famous snack is the lu rou huo shao, in which chopped, spiced donkey meat is mixed with fresh green chilli and stuffed into a layered pastry - its name is  sometimes translated into English as a "donkey burger".
Donkey meat may also be made into soups and stews.
As the Chinese middle classes have grown richer, demand for donkey gelatin and other tonic delicacies has soared - in January the New York Times reported that a shortage of donkey gelatin had encouraged a boom in imitation products.
The ban also covers the hide  of horses and camels, Burkina Faso's director for public veterinary health Adama Maiga told the AFP news agency.
The government would also be "regulating" the slaughter of the animals, he added.
Last month, an abattoir set up by French and Chinese businessmen in a village on the outskirts of the capital, Ouagadougou, was ransacked by residents protesting against the stench coming out of it and the air, water and land pollution it was causing, AFP reports.
Burkina Faso, a poor West African state, has about 1.4 million donkeys.
Exports of the skin rose from 1,000 in the first quarter of 2015 to more than 18,000 in the last quarter, the Burkinabe authorities say.
Nearly 65,000 skins were exported in the first six months of this year, mostly to China.
Its donkey population has been dwindling because of low fertility rates and the long rearing period, causing a a shortage of hides used to produce a traditional medicine known as "ejiao", according to a report in January in the China Daily newspaper.
The medicine is taken mostly by women who suffer from anaemia, dry coughs or dizziness.

The development will be an annex to the school for three to 18-year-olds in Ystalyfera.
Built on the site of the former Sandfields Comprehensive, the secondary school will have 650 places.
Education Secretary Kirsty Williams said: "This money will address the region's growing demand for Welsh medium education."
The Welsh Government has awarded £10.69m towards the project as part of the 21st Century Schools programme which aims to update school buildings across Wales.
Burkina Faso has banned the export of donkey skins as a sharp increase in sales to Asia is threatening the animal's population.
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Money for a new £17m Welsh medium school campus in Neath Port Talbot has been approved by the Welsh Government.