Hello from JustAnswer.
There are a couple of possible causes of burning when you breathe out.
The first condition that I would be concerned about would be some irritation of the airways from acid reflux. Acid reflux will often cause heartburn, but there are some people that do not get heartburn and instead present with symptoms from irritation of the airways or throat. The stomach acid can cause a chemical burn of the airways, and this may be felt as a burning pain when you breathe.
An infection of the airways, called a bronchitis, also can cause irritation of the airways. It is not commonly described as a burning sensation, but it can do so. Bronchitis is very common, so an unusual presentation of a common problem could still be the cause. It is also possible that there could be inflammation in the airways as a primary problem, such as from allergies, but this would be even less likely to cause a sensation described as burning.
As long as you are not having other symptoms and particularly not having any shortness of breath, it would be OK to try some home interventions. At this point, it would be reasonable to try an over the counter medicine to help suppress stomach acid, such as ranitidine or omeprazole. It also may help to take an antihistamine, such as cetirizine or loratidine.
If I can provide any additional information, please let me know.