The planet I'm thinking of is of similar size and distance from its star relative to Earth, but it lacks oceans, instead consisting of one massive continent with a huge mountain range (reminiscent of the Andes) running east-to-west around the equator, forming an almost perfect mountainous "ring" around it. To the south of this mountain range is a vast tropical rainforest, and to the north is a massive desert. The mountain range itself tends to chop up air masses and creates microclimates. How feasible would a world like this be, and what changes to the climate I already have set for it would I need to make to make it be able to support human civilization with no oceans?
On Earth, oceans exist because rainwater runoff collects into large pools called "oceans". Oceans are at the lowest point on earth, so the water has no where else to go, and it stays there until it evaporates.
Ways to eliminate oceans:
Bury the oceans. Earth's mantle contains at least as much water as all of Earth's oceans, dissolved in various minerals. On your planet, in the northern hemisphere, the crust is very porous, and the ocean water sinks into the mantle at a high rate. If the mantle is able to retain all of it, then most of the water will be lost off of the surface and be trapped deep underground. In other words, the water table will start tens of thousands of meters under the surface. This will result in a great northern desert. Vapor water might re-enter the surface water-cycle from volcanoes.
Put the oceans in holes. In this scenario, you will still have oceans, but they will take up very little surface area. The planet has many very deep fissures (much deeper and much more frequent than the Marianas Trench). The total volume of these fissures/trenches is such that the oceans can entirely fit inside of them. If you look at a map of the planet, it will (misleadingly) look like the planet has almost no water, because the planet will be almost entirely land area.
Freeze the oceans. If the planet's star is very weak, you could force the planet to be tidally locked to the star. The near side ("south" side) would be temperate, and capable of supporting a water-cycle and rain-forests. The far side ("north" side) would be a frozen desert, like the Antarctic Desert.
You can't, unfortunately.
Without surface water you won't have a sustainable hydrologic cycle to keep life flourishing in the variety and density of a tropical jungle. Water will disappear into the crust beyond the reach of any root system (the Earth's crust is estimated to contain as much water as all the oceans, if not significantly more) in greater volume than it can be replaced through volcanic outgassing transpiration and evapouration. You may have noticed the Amazon, Congo, and other jungles are restricted to certain parts of the world: where prevailing tropical winds bring moist sea air inland, not just any coast at any latitude, and not just anywhere on land.
To the south of this mountain range is a vast tropical rainforest, and to the north is a massive desert.
An Earthlike world that isn't tidally locked cannot have hemisphere-wide climates, due to atmospheric circulation patterns and unequal heating. Handwaving the lack of water for a moment, you would still have bands of different climates where convection cells rise and fall. But without any surface water, you would have a planet-wide desert (cold at the poles and elevation, hot between, but desert).
Here's the problem: you have posited that there is free water, and plenty of it ('rainforest'). So, where is that water going? Either you have lots of lakes, or they get bigger and are "oceans." In either case, over time these water bodies will get salty just as they did, and do, on Earth.
Next: do you want this planet to be habitable when humans come to visit, or are you positing carbon-based life developing on this planet? It doesn't really matter except for the possible difference in atmospheric composition: sufficient oxygen, for example.
In the book "The Stone God Awakens" by Philip José Farmer, humanity engineers trees to provide living quarters. Over a vast amount of time, humanity is gone and the trees have supersized and taken over almost all the planet.
If you have the seas becomes a glorified mangrove beneath the root of supersized plants, you could create the effect you're looking for. The seas are still there but buried under a mile of roots and leaf litter.
Seriously, lack of oceans makes it extremely unlikely that your humans and the trees in the rain forest evolved on this planet. It is simpler to assume they were imported from somewhere else by people capable of interstellar travel and artificially creating an environment capable of supporting them.
In this scenario the high dividing mountains would have been created artificially. Either by artificial volcanism or by dropping lots of space rubble on the equator. Either case this would have been used to add volatiles to the surface.
Moisture needed to maintain the rain forests would be provided by terraforming machines pumping up moisture from the mantle or other deep reservoirs. These were either only built or only work on one hemisphere with the other having dried out as water has been absorbed by the ground or lost to space. In the latter case the atmosphere in general would be artificially maintained at reasonable pressure with provided nitrogen and oxygen crossing the mountains but water coming down as rain due to the mountains
Rain cycle and water storage
I haven't seen any answers discuss the possibility of having a water cycle just from the evaporation of water in the plants in the rainforest. The rainforests on Earth evaporate huge amounts of water just from the plants' surface area. This evaporation can then fuel cloud formation, resulting in the necessary rain.
Combining this with a porous surface layer beneath the forest that can hold the bulk of the water, you might not need the 'open' surface water of oceans. A clay-like ground layer beneath the porous layer could then keep the groundwater from seeping deeper into the crust.
Equatorial mountain range
Coming from an astrophysical point of view, I think your equatorial mountain range is totally possible, even without the plate tectonics! In the formation of this planet, there might have been a ring of debris around it that has accreted onto it over time, leaving a perfect ring of mountains. Plate tectonics would probably even ruin this scenario, since the ring would be broken up and become dispersed or even disappear entirely.
Desert above, forest below
The formation scenario for this sharp ecological divide from south to west is a difficult one. If such a flourishing forest would arise in the south, why would the north be a wasteland? I can imagine a couple of scenarios.
1) During formation of the planet, two major (protoplanetary) bodies collided, and the material of them did not mix very much. This would give the possibility that the one side of the resulting planet has a surface with unfavorable conditions for complex life to evolve, while the other side has good conditions.
2) An asteroid impact in the north erased most life on that side (in a long lasting winter due to obscured skies) while the mountain range kept the impact of this impact (pun not intended) on the southern hemisphere small.
Our solar system has far more planets without (water) oceans than ones with oceans. The hard part is making it habitable as well, and the request for rainforests. So long as it doesn't need to be too hard-sciency, here's my go for an explanation.
There used to be more water, there used to be oceans. A few centuries ago, some event removed a significant amount of water from the environment. Maybe it was an asteroid impact, maybe enough soil-mass was lost and the water simply sank underground. For whatever reason, there is now less above-ground water than their used to be.
This means that the land has giant salt-flats where the oceans used to be. The rivers still run into the sea-beds, but the sun evaporates the surface water as it spreads out on the river delta's. Thus you can have rivers, maybe even lakes in the mountains, but on the old sea beds the water is just gone.
The old seas are needed to create large lifeless (salty) areas where the water can evaporate and provide the hydrological cycle. Over time, the salts will gradually wash downhill in the sea beds after floods and the sea beds will become habitable and the hydrological cycle will change. But for a few centuries, perhaps, you can have rainforests and no oceans.
Reality check: the loss of water from the atmosphere would probably cool the planet significantly which would probably result in surface ice rather than desert, but, eh. In my mind this sort of environment can probably be handwaved - you may not need to provide a detailed geological account of how it occured. In Dune, Frank Herbert never explicitly stated why the planet Arrakis was a desert planet, it simply was and few readers questioned how possible it is.
You might be able to design a tidally locked planet around a red dwarf with properties like this although you'd have to have oceans on the back side underneath a deep layer of ice and at least somewhere on the side facing the sun with some open ocean. That could explain in part your rain shadow effect. If you have one large Pangea like continent set up kind of as follows: from South to North or reversed:
South<-- Ice here <-- small surface ocean<--Expansive Desert <-- Continent dividing Mountain Range <---Tropical Forests <--- More ocean << Ice here <- North < Dark side hemisphere is ocean covered in ice.
The mountain range would "bisect" the continent and winds coming from the dark side across the northern ocean would bring moisture that would be dumped on the tropical rain forest areas. Problem: Polar Winds coming from dark side might be extremely dry. For rain you need a source of water that can go into the atmosphere. You might get some from sublimation of ice from the dark side and then water would flow back to the dark side in rivers to freeze sublimate and rain again. Look up papers for rain shadow effects and tidally locked worlds and you might find a suitable solution.
Ideally however if you have plant life evolving on your planet I would say you need a source for your water cycle and that means a large source of water that cycles from one form either solid to vapor or liquid to vapor to allow for rain shadow effect, rainfall and flow back to whatever the large source of water is.
Completely oceanless might be feasible but I think you'll need some large water body that is intermediate between your source of rain water and where it ends up in the cycle.
Another possibility might be some kind of groundwater but that seems a bit far-fetched to me. Anyway hope this helps.