Humans have rights and obligations. The obligation connected to certain rights amounts to the fact that other people's rights should be respected.
Animals (should) have rights, but what about their obligations?
Apart from how you view the content of animal rights, animal rights are in fact human rights. These are rights afforded to humans in order to set boundaries for others in their treatment of animals.
For this very reason, animals do not have obligations. The obligations lie with the humans who wish to set boundaries for others and for those fellow men who deprive animals of their freedom.
In our time, space on Earth for Nature has become so limited, that you could say that control of all space on Earth is in the hands of humans and, consequently the responsibility for it. Just one or two centuries ago, the situation was entirely different. Before that time you could retire from society to live in Nature in Africa or the Wild West and conduct business there that had no jurisprudence whatsoever.
Luckily, those times have passed, although that period is still romanticized in movies with cowboys and indians.
Now the State is responsible for care of animals in border areas that have been wounded or that could cause possible harm to humans, livestock and property.
Animals wounded in traffic are simply killed by the authorities (police). They no longer need to provide funds for transport of the animals to animal hospitals.
Over and done with.
In order to change this irresponsible fundamental attitude of politics and government, an integral awareness change is needed, which is endorsed, in principal, by every citizen. To give this change a chance of success, a vision is required, which is both simple and forceful. To put this into words, we would need to go to the level of the development of civilization. This development is characterized by leaps in degrees of freedom. Each living creature can only evolve with a chance of success, when the following step affords more freedom.
Freedom is a means and an end to arriving at more quality of life.
There is so much to be said about this. This article began with the question as to whether animals have obligations. The answer has been provided. We now have to wait for development of civilization to mature, a development only we humans can initiate and maintain.