I'd like to give you a happy fun thought experiment to chew on.
It's 2016. And it's been a bad year. Let's imagine that it's about to get infinitely worse for everyone, and by December 31st, 2016, the human species is extinct. Cause: something minimally disruptive to the rest of the biosphere. (A very tightly targeted human-specific military bioweapon gets out and proves to be unexpectedly deadly: say, an IL-4 expressing poxvirus that goes above and beyond.)
Earth abides, of course, and without humans life goes on.
Let's ignore the immediate aftermath (1-1000 years). Nuclear reactors scram automatically, grids shut down, there are various nasty industrial accidents from unattended plant, and then the atmospheric carbon pulse continues and is joined by large-scale outgassing from the Siberian tundra and possibly a crash in hard-shelled ocean dwelling species due to acidification. Global mean temperatures rise by roughly 4 degrees celsius (hey, we're not pumping any more CO2 out!) leading to considerably worse weather events and various ecosystem changes: the ongoing mass extinction event continues to coast on momentum during this period as more specialized species fail to find new niches.
(Noteworthy exceptions: rats, cats, dogs, pigeons, grasses, and probably goats are almost everywhere, thanks to human activity. So are a bunch of other species, but sheep, cattle and horses are less prepared to go feral if you abruptly remove all human intervention. Nevertheless, there will probably be new ovine and aurochs subspecies in the wild in places far from their original point of domestication by the end of the 1KYa marker.)
So, let's look to the long term.
Vertebrate life on Earth dates back roughly 525 million years. Our sun is gradually brightening, and over the next 200 MYa - 2GYa period the increased UV flux will split water vapour in the upper atmosphere into hydroxyl radicals and hydrogen ions—and the latter are sufficiently fast-moving to be lost into space via the solar wind. Over time this will dehydrate the surface and then the upper lithosphere, baking the planet into a cooler, more massive version of Venus. It may take a couple of billion years for the last life forms to die out, but extinction beckons.
However, between the end of the human epoch and the end of a biosphere capable of supporting vertebrate life there stretches a span of time considerably greater than the span separating us today from the first dinosaurs, 241-243 MYa ago.
Continental drift is going to play a role in large-scale evolutionary trends; geologists speculate that between 50 and 200 MYa from now we can expect the Arctic Ocean and Caribbean Sea to close, resulting in a fusion of the Americas and Asia centered around the north pole into a new supercontinent, Amasia. An alternate model proposes the formation of Pangaea Ultima around 250MYa hence. It's very unclear which supercontinent model will prevail, but we do know that roughly every 250-500MYa the Earth's continental plates drift into a single mega-continent configuration, and then subsequently drift apart.
Megacontinents have implications for life because their interiors tend to be arid and often cold—they correlate with large scale glaciation and the interiors are not hospitable. Formation of supercontinents also seems to correlate with large-scale changes in atmospheric oxygen levels. It's also worth noting that the maximum size of vertebrate species increases with the size of the biome accessible to them—the converse effect (island dwarfism) of reduction in size of large animals living in small/constrained areas, such as islands, is clear, and it is noteworthy that the sauropods and titanosaurs (the largest land animals ever) flourished between 240 and 66 MYa ago, originating on the supercontinent Pangaea and continuing as it split into the large continents of Laurasia (which later split into North America and Eurasia) and Gondwana (Africa, Antarctica, South America, Australia, Arabia, and the Indian subcontinent): in other words, huge land masses.
So: 50-200MYa hence, we can expect supercontinents to emerge, and with them, really big land animals. Possibly countering this is the issue of atmospheric partial pressure of oxygen; if it hits 30% (it's currently around 21%) even waterlogged organic tissue will burn, so forest fires resulting from lightning strikes can be expected to devastate surface level ecosystems. (Dinosaurs proliferated in a less oxygenated environment because they had highly efficient lungs, like birds; no surprise there because birds are, in effect, our surviving micro-dinosaurian neighbours.)
But what else is going on ...?
Over the past 200-odd MYa we've seen a couple of arms races driving vertebrate evolution. The first of these was the development of venom; for example, in snakes, venom originated roughly 170MYa ago; tetrodotoxin mutualism (in puffer fish, blue-ringed octopi, and other species) is probably more recent. Many species not routinely thought of as being venomous may indeed poison their prey by biting; for example, the common house cat is well-known for "cat scratch fever" and for bite wounds becoming infected—this may play some role in predation, and I'd speculate that as resistance to toxins emerges over time, so too will more potent venoms be selected for (both by predators, and by edible/prey species that rely on poison to injure or kill predators).
The other arms race is, of course, theory of mind. A predator that can model the behaviour of its prey species is one that can hunt more effectively; and a prey species that can anticipate likely predator strategies is one that can avoid being eaten. We can't know much about dinosaur hunting strategies (and, based on observations of contemporary birds, it would be very unwise to assume that small brains mean low intelligence), but those contemporary species that put lots of metabolic energy into adapting limbs and dentition for killing seem to be less reliant on general intelligence (specialized, saber-toothed big cats have a smaller encephalization quotient than more generalist feline species). Either way, encephalization quotient seems to be slowly increasing over geological time, and we can suppose that predator activities may also be becoming gradually more complex and sophisticated.
So what is the world going to look like in 50MYa?
In the sea, I'm not sure the bony fish are going to recover from what we've done to the ocean food chains any time soon: they may be replaced by cephalopodia (and jellyfish at the invertebrate end). If so, expect some radical new cephalopod forms to show up as they radiate to occupy the niches vacated by the big fish. (Cephalopod plankton filter-feeders like basking sharks? Giant, aggressive fast-swimming squid replacing schools of tuna? Who knows?)
Birds (dinosaurs) and mammals are going to survive and be the main megafauna on land. (Big flightless birds, less so—the need to lay eggs, and the limit on egg size imposed by needing a large surface area to volume ratio for O2/CO2 gas diffusion, puts them at a disadvantage when trying to raise a clutch in an environment full of egg-stealing mammals; it's worth noting that the extinction of the Phorusrhacidae (Terror Birds) in South America coincided with the formation of the Isthmus of Panama and the arrival of mammalian predators.) Supercontinents mean scope for larger apex herbivores, in the extreme scaling up to the size of titanosaurs: a higher atmospheric pO2 may allow mammals to become nearly as large as the larger dinosaurs, although poorer lung performance works against that. But supercontinents also mean poor weather and possible large scale glaciation events.
I'd expect to see more and more species employing venom to hunt their prey, and toxins (often obtained via mutualism/symbiosis from bacteria, like TTX today) to poison their predators: if you see something harmless, fearless, and furry, like a skunk, you should probably avoid eating it if you don't want to die in convulsions.
And we're going to see theory of mind everywhere. Think in terms of bears building fish traps in rivers and pit traps on land. Wolf-analogues coordinating their hunting drives by stationing individuals at high points to relay signals. Raccoons ... shudder.
What am I missing?