The magic skunk-smell-be-gone recipe:
In a plastic bucket, mix well the following ingredients:
1 quart of 3% Hydrogen Peroxide
1/4 cup of baking soda
1 to 2 teaspoons liquid soap
First thing you want to do is get as much of the skunk goo off of you as you can. Using paper towel, tissues, or a rag you can throw away, dab the spray off your skin. Rinsing ...
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 ...