A common adjective would be ***worn-out***. (as in *worn-out donkey*).

> (Of a person or animal) extremely tired; exhausted:

> <sup>http://www.oxforddictionaries.com/us/definition/american_english/worn-out</sup>

Note: *Worn-out horse* is more common than *worn-out donkey*. Personally, I heard or read *worn-our horse* more than *worn-out donkey* also.

Below [Ngram comparison][1] ( between *worn-out horse, worn-out donkey, worn-out mule*) might give an idea also but it doesn't prove how widely accepted the terms are in general:

![Google Books Ngram showing *worn-out horse* being more prevalent than *worn-out donkey* and *worn-out mule*][2]

Example:

> Under fire, he "hugged the ground more than once," fatigue knocked him out, and sometimes he collapsed like "a worn-out horse."

> *[Under the Bombs: The German Home Front, 1942-1945 By Earl Ray Beck (1986)]*


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As an additional information, it is more common to use "as tired as an <_animal noun_>" phrase without using an adjective before the animal noun. Though, it seems like this usage has fallen dramatically. *Dog* is still more common than *donkey* or *horse*. 

Below [NGram comparison][3] might give an idea also:

![Google Books Ngram showing use of *as tired as a dog*, *as tired as a horse*, and similar phrases for other animals][4]


As Mari-Lou A suggested in the comments, *dog-tired* is more common than any of these as an animal-related term to indicate exhaustion.


  [1]: https://books.google.com/ngrams/graph?content=worn-out%20horse%2Cworn-out%20donkey%2Cworn-out%20mule&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cworn%20-%20out%20horse%3B%2Cc0%3B.t1%3B%2Cworn%20-%20out%20donkey%3B%2Cc0%3B.t1%3B%2Cworn%20-%20out%20mule%3B%2Cc0
  [2]: https://i.sstatic.net/avBVD.png
  [3]: https://books.google.com/ngrams/graph?content=as%20tired%20as%20a%20*%2C&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t2%3B%2Cas%20tired%20as%20a%20*%3B%2Cc0%3B%2Cs0%3B%3Bas%20tired%20as%20a%20dog%3B%2Cc0%3B%3Bas%20tired%20as%20a%20pack%3B%2Cc0%3B%3Bas%20tired%20as%20a%20donkey%3B%2Cc0%3B%3Bas%20tired%20as%20a%20man%3B%2Cc0%3B%3Bas%20tired%20as%20a%20horse%3B%2Cc0
  [4]: https://i.sstatic.net/5YTeh.png