A young Tyrannosaurus Rex would be eaten
If you've ever looked at collections of dinosaur fossils, you might wonder about why there are really big predators (like T. Rex) and really small ones (like Velociraptor, actually the size of a big turkey) but nothing in between. That is because Tyrannosaurus was the mid-sized predator.
Above is the growth chart for T. Rex. Notice that T. Rex takes about 10 years to grow to the size of grizzly bear; and won't reach full size until it is more like 20. A lion, on the other hand, rarely even lives that long in the wild. Wikipedia gives the average age of a male at 10-14 years. Meanwhile a cub hits sexual maturity in 2-3 years.
So there is the problem. A young tyrannosaur would have to survive for 5 years at least avoiding predators. Just looking at Costa Rica, a baby tyrannosaur would be avoiding otters, racoons, weasels, foxes, grison, tayra, skunks; a predator for every habitat. He wouldn't be safe in the trees, on the ground, or in the water. And then if this tyrannosaur survived to 10kg, he is still potential food for a cougar or jaguar.
You might think, well baby T. Rex survived just fine in the Mesozoic, why is so hard about now? The difference is endothermy in all the creatures that baby T. Rex would be competing with. Warm blooded creatures are much more active than cold blooded. In the modern world, T. Rex isn't keeping away from lizards and snakes and proto-mammals; it is avoiding high metabolism predators that have to eat half their body weight each day. The whole paradigm of 'have a lot of eggs and hope for the best' is literally extinct in today's world for large animals. If you want a small baby to survive to a big animal, you need to take care of your offspring, and you need to make it grow faster. You need to feed it milk, which means that you need to be a mammal. Exceptions in modern ecosystems are strictly limited.
A big Tyrannosaurs Rex would starve
Can dinosaurs run? Not very fast, probably, given their sizes. In the olden days of the Mesozoic, the way to protect yourself was to have lots of offspring, and grow to massive size. After all, we have just gone over how long it takes a T. Rex to get to huge size. Huge T. Rex are rare! Most of the T. Rex that your average hadrosaur encounters on a day to day basis are going to be much smaller; too small, in fact to bother a 6 ton hadrosaur. Herbivores evolved for size (like sauropods) or toughness (ceratopsian or ankylosaurs). Its pretty hard to argue that any of those three dinosaurs was designed to outrun a predator.
Today, if you are a large herbivore, you protect yourself by running.
Now, lets address the T. Rex life cycle. As a newly hatched dino, the T. rex can probably find enough to eat with lizards and rodents and what have you. But as it gets bigger, it starts to have a problem. What does a 25 kg T. Rex eat? Its a bit big to be going after mice. It can't climb trees like the comparably sized ocelot. It can't run as fast (or more importantly, as long) as a coyote to track down small deer in a forest. The best it can do is scavenge.
What does a 100 kg tyrannosaur eat? There is no way it is going chase down a deer. Or even a peccary through thick brush. Those creatures are both designed to evade jaguars. A jaguar will be stealthier, due to its low profile, and much more agile in a chase through the forest, due to having four legs.
Lets say that this Tyrannosaur somehow makes it to 500 kg....now what does it eat? It isn't fast enough to catch anything anymore. There is no thing slow enough for it to overpower with size. Maybe 10,000 years ago it would feast on ground sloth and elephants, but those aren't around any more.
The closest thing to proof
When North America and South America come together, a few million years ago, the continents exchanged predators. From the North came the cats, like sabretooths and cougars and jaguars. From the South came the Terror Birds. Weighting up to 150 kgs, they are in many ways an analogue of the juvenile T. Rex in the 5-10 year old range. They could run fast, had excellent vision, and powerful beaks and claws.
They are also extinct. The big cats are not.
I tried to make this as short as possible. It is quite impossible for me to cite every little statement in here, so you will have to do your own research to see if you agree with me. The bottom line is that the way animals were designed and that the ecosystem interacted in the Mesozoic was radically different from how it works today.
In the Mesozoic, there were tons of immature animals of every size, enough to provide a continuous buffet of meals in all weight classes. Today, mammals power grow their children to full size in 2 or three years; there just aren't that many half-sized deer or what have you to catch.
Today, animals are built for the chase. Utilizing the mammals warm blood, every good prey item from rabbits to antelope to horses is designed to run away, fast, and with endurance. T. Rex simply isn't built like that. Even if it were a stealthy stalker (which is dubious), it would not have the agility on two legs to catch the deer or wild pigs it would need.
No, the age of the T. Rex is firmly over.
Question says 'modern ecosystem.' The possibility of T. Rex gorging on cattle is right there, but since the question is whether it would survive in the wild, I'd say it is a resounding no.