I can only assume that the intended meaning of “inheriting” the earth in the wake of humans isn’t a general survival standpoint, but a question of what variety of animal has the best chance of achieving sentience, and a society that allows the species to reach a global scale, and the ability to effect their environment comparable to our own.
If it’s just a matter of survival, nobody’s going to inherit anything from humans, because arthropods like ants and spiders already have us beat. They exist in numbers far superior to our own, and have been around for a remarkably long time, and require considerably less than humans do to survive. Being tiny will do that for ya’. So I can’t imagine that’s what we’re discussing.
Humans have a wonderful set of traits going for them that enabled them to alter their environment on such a grand scale, and reach such significant numbers. Mental acuity is an obvious component, but social skills and abilities (particularly communication) are probably even more important than that. And, of course, they need to have some appendage with dexterity comparable to that of a human hand, which is a piece of art from an anatomical standpoint. All of these are a result of our primate nature.
Mental acuity is a vital one, and perhaps more importantly, a great stress on learned behavior rather than instinct, and the ability to keep learning throughout one’s life. Primates, particularly hominids (in this context, not referring to bipedal ancestors of modern humans, but members of the family Hominidae - the “great apes”), tend to have a high ratio when comparing brain mass to body mass. They certainly learn well.
Humans in particular display the lovable trait of neoteny in spades - a tendency to display juvenile traits even in adulthood. The skull and face of modern humans reflect the appearance of juvenile great apes (women particularly). But physical traits retained into adulthood are less important than mental ones in this case - for many animals, almost all of the learning they do is done in childhood, and very little else is gathered in maturity. Information maybe, but learning new skills and concepts isn’t as easy. Humans, neotenous scamps that they are, keep learning throughout their life.
Social structure is another important one. Neoteny helps here again - the malleable nature of the mind aids in accepting new people, new relationships throughout one’s life. And far before primates are intelligent, they are social. Not all are gregarious, but many are, and if nothing else the bond between mother and child is significant in most primates. Our closest living relatives - gorillas, common chimpanzees and bonobos - are all very social, and certainly gregarious. An interesting array of solving social conflicts are expressed in those three species as well.
Our own dim views to the contrary, humans do in fact manage to live together in a remarkably peaceful manner. It is not always fair, right, and certainly not always egalitarian. But compared to other species, humans fare quite well when large numbers are forced into close quarters with limited resources. Consider the behavior of a great number of rats placed into a confined space, by comparison. We also manage to work within a great number of social groups throughout our lives, not just our family units, which is odd indeed, and vital to the development of culture and all the advantages that go with it. It’s been suggested (keeping in mind that you’ll find studies and experts to support anything; it’s seeing that same conclusion reached by other studies and individuals that starts to lend true authority to an idea) that Neanderthals suffered from a more insular social structure, and that this inhibited the spread of ideas and technology that modern humans engage in. Of course, it would also prevent them from banding together in larger groups for defensive - or offensive - purposes. They’re mostly theories, so…grain of salt and all that.
Aiding both mental and social development is the extremely long period of maturation that primates endure. One would think a long childhood to be of little use, but even as neotenous as we are, we still learn fastest when we are children. That protracted childhood lends itself to the accumulation of knowledge and skills; learned behavior is the great strength of humanity.
And then there’s the issue of physical apparatus. Fate conspired to create perfection itself for this in humans. It’s in our background - primates are known for having a remarkably…“primitive” skeletal structure. That is, primates have specialized anatomically far less than other animals in most regards, sticking instead with the flexibility expressed in our most ancient mammalian ancestors. The arboreal lifestyle of most primates benefited greatly from that flexibility. The dexterity of primate hands and feet are evidence of this, but also of importance is our very flexible shoulder - the same range of motion that allows for brachiation allows for a wide variety of movements for tools.
And then, of course, there’s being bipedal. Though the hands and feet of other primates can be very nimble indeed, they are hindered by their more basic use as locomotive appendages. It’s not just a question of their hands and feet being busy moving them along most of the time, but the adaptations those digits have to support the weight and endure the impact of moving their owner. Human hands are remarkable, but certainly not sturdy - how much of that dexterity and fine precision would be lost if they also had to be adapted for quadrupedal movement?
That said, the reason for human ancestors shifting to an exclusively bipedal lifestyle is a mystery. It’s remarkably disadvantageous, when one considers the typical lifestyle of a primate. A common theory has been that a shift in the climate caused forests to give way to grasslands, on which a bipedal stance would be useful for the higher vantage point it offers. The problem is, in recent years fossils have been discovered that show bipedal apes emerging six million years back, when their African environment would still have been quite lush. Locomotive advantages were nil, at least at first - the change in anatomy that allows for upright movement cripples the ability to climb and maneuver in trees. What advantage would this establish in a forest? True tool use - in the sense of fashioning tools deliberately, and using them frequently - did not emerge until 2.5 million years ago. So that’s no advantage. Some have suggested that the upright stance could be used to carry young or food, but the young of most primates have little trouble holding onto their mother, and the advantage of transporting food is debatable. Even movement on the ground would have been slow and inefficient at first, as the body wouldn’t be as well adapted as ours is now - and the modern human skeleton is quite flawed indeed.
Whatever advantage bipedalism conferred (and there must have been something, particularly considering that primates further removed from our own lineage, with no living descendants, also experimented with bipedalism around 6 million years ago) in the short term eventually allowed for the enormous advantages it has given us now.
The problem then, is in finding a variety of animal that can replicate this set of traits. Felids are a poor choice, as they have specialized thoroughly to hunting, and rely far more on instinct than learned behavior, and relatively poor social ability. Wolves have the potential in the sense of relying far more on learned behavior and being a part of the highly social canid clan, and can even express high degrees of neoteny (the trait that most distinguishes dogs from wolves) but they are hampered physically by the highly specialized form they have. Wolves are designed to run - they are not flexible, and a wide range of movement is in fact a disadvantage for them. Their limbs, digits, shoulders, and other adaptations are all quite locked into this lifestyle. Arthropods suffer from a very clear size restriction (for a variety of reasons, including the effect gravity has on exoskeletons at a large size, and the length of time it would take to grow larger exoskeletons), and have very limited mental abilities. Cetaceans are highly intelligent and social, but they also have highly specialized physical forms that do not lend themselves to tool use and creation. Birds once again suffer from the lack of physical flexibility; their wings are too specialized to even hope to perform any proper tool creation or use, even if they do have a lovely range of movement. Their beaks display great ability in this regard, but that still has significant limitations, and inhibits verbal communication.
Primates are obviously a good choice, but I doubt that other primates would survive any catastrophe significant enough to kill off humans. Some primates, maybe - but certainly not the great apes, or the other primates most closely related to us.
It’s possible some variety of animal could replicate these abilities in some fashion, but it would be in a very different manner than our own, and so it is difficult to imagine how it would work.