TLDR: It has happened once to a dog and it is possible that it could happen to birds but I can find no instances of it happening.
In rare cases, skunk spray can have more effects than just smell on a dog.
In rare instances, Heinz body anemia, methemoglobinemia, and hemoglobinuria may occur a few hours to 24 hours after exposure (ASPCA Animal Poison Control Center Antox: Unpublished data, 2011).5 In these cases, the thiols in the skunk spray cause oxidative damage to hemoglobin.
...
Although there are no reports of a cat developing methemoglobinemia from skunk spray, feline red blood cells are more sensitive to oxidative damage than are the red blood cells of other species.
The dog escaped from the house and when the owner found the dog the next morning, the dog smelled strongly of skunk spray and was tremoring. The dog was brought to an emergency clinic more than 12 hours later. Nearly 100% of the red blood cells studied contained Heinz bodies. Results of laboratory testing confirmed methemoglobinemia. The dog had a seizure and died. The owner requested cremation and did not authorize the release of histopathologic and other diagnostic findings. To our knowledge, this is the only death related to a skunk spray in a dog.
Skunk spray toxicosis: An odiferous tale (Emphasis mine)
There was also a study done on this,
BACKGROUND:
A captive Red Panda developed a regenerative anemia with Heinz bodies after being sprayed by a skunk. A definite cause-and-effect relationship between skunk musk and oxidative erythrocyte damage has not been reported, but it was suspected in one reported case of a dog with Heinz body hemolytic anemia.
...
CONCLUSIONS:
In vitro, skunk musk causes Heinz body and methemoglobin formation in canine, feline, and Red Panda RBC, supporting the clinical association between Heinz body hemolytic anemia and skunk spray exposure.
Skunk musk causes methemoglobin and Heinz body formation in vitro.
As far as birds go, it is possible for them to get Heinz-body hemolytic anemia from ingesting crude oil but I found nothing on them being sprayed by skunks.
