Wesley J. Smith has written about a case in Argentina in which an orangutan named Sandra has been declared by the court to be a non-human person. This ruling essentially would grant Sandra her freedom because it is unethical to hold people captive unlawfully.
The BBC has even written about litigation in the United States to try to get Tommy, a chimpanzee living in captivity, recognized as a "legal person."
Peter Singer and Michael Tooley are two atheist philosophers who have long supported this notion that certain animals, like chimpanzees and dolphins, ought to be considered persons and human infants, embryos, fetuses, and the severely disabled ought not to be. Now it seems that fight has gotten bigger, even being won in at least one part of the world.
This line of thinking is atheist at its heart. If we are just the product of random mutations, a blind, naturalistic process of evolution, then really we are no more special than any other animal species out there.  This isn't just my interpretation, either. In Matter and Consciousness, Paul Churchland wrote: "The important point about the standard evolutionary story is that the human species and all of its features are the wholly physical outcome of a purely physical process...We are creatures of matter. We should learn to live with that fact."
In fact, Singer and Tooley both (in Practical Ethics and Abortion and Infanticide, respectively) have argued that this idea that humans are intrinsically valuable is a religious idea, and we need to do away with these "antiquated" notions. The only reason we now view infanticide as wrong is because Christian morality has permeated Western civilization.
So if you are going to argue that certain animals deserve personhood, one of two things must happen:
1) You raise those animals to the status of human beings. So if Tommy and Sandra are persons, and infants are not, that means that killing a chimpanzee is more serious than killing a human infant. If you "murder" a chimpanzee, you deserve to be locked away in jail for life or executed.  If you kill an infant, you don't. At most, you would be guilty of a property crime, if you killed the infant against the parents' wishes. This is a highly counterintuitive idea.
2) You lower humans to the status of mere animals. After all, if we're no intrinsically different than animals, then it's really not seriously wrong to kill another human being. This is also highly counterintuitive, since we have very strong intuition that killing any human being, especially ones that are younger and more vulnerable than we are, is seriously wrong.
The bottom line is chimpanzees don't deserve personhood status. Personhood is not some arbitrary idea that we can just ascribe to entities and consider them "one of us." Being a person is inherent in being the kind of entity that you are (in our case, a human being made in God's image, or to state it in a secular way, an entity with an inherent nature as rational agents). Those who believe you are a person based on the functions you currently perform are guilty of a simple confusion: confusing being a person with acting as a person. As you must be a human before you can develop human parts, so you must be a person before you can develop personal properties.
I do not believe that animals are rights bearing entities. That's not to say that we can mistreat them. They are still entities that feel pain, and we should respect that. And a human being who mistreats an animal is at risk of becoming animal-like, themselves, becoming desensitized to pain in others. However, even if you believe they are rights bearing entities, you can ascribe rights to them without ascribing personhood to them. But what we also have to understand is that with rights comes duties. I have a right to live. This means that I am also obligated to respect everyone else's right to live. Apes cannot understand or abide by any obligations.
Sandra and Tommy are both blissfully unaware of these court proceedings. In fact, they don't care one iota about whether or not you consider them persons. This is an important difference between apes and humans. Apes may be highly intelligent -- but only when compared to other animal species (this is often lost in the animal rights debate). Apes are not very intelligent when compared to human beings. No ape will ever write like Tolstoy, or paint like Michelangelo, or compose music like Bach, or fly other apes to the Moon. And while apes may be able to use certain rudimentary tools -- that's all it is, a rudimentary tool. No ape will ever open a hardware store for the carpentry needs of other apes. And while infants may not yet be able to understand these rights and obligations, they will. And that's the crux of the matter. Human embryos/fetuses/infants will naturally develop these abilities, whereas apes never will.
So trying to ascribe personhood status to lower animals is unnecessary and makes a mockery of human dignity. Animals have been part of the ecosystem for a long time. Animals kill each other, protect each other, copulate, and do all manner of things without human help and will continue to do so without human intervention and without caring a bit about personhood or what it is. There is just no reason to ascribe personhood to animals. The only possible reason would be to ensure that humans don't mistreat animals or cause them to go extinct. But ascribing them personhood status is not necessary for that, either.
 I'm not wanting to get into a debate about evolution. It's certainly possible that God used evolution as the mechanism by which to create humanity. The operative idea here is if naturalistic evolution is true, this is the idea that would follow.
 Again, the debate regarding capital punishment is beyond the scope of this article. This idea here is that whatever the penalty for murder is, that's what you deserve if you "murder" a chimpanzee.