Oil is pretty ubiquitous in our lives, right? All the kayaks in the Willamette River protesting the movement of a Shell drilling platform in 2015... were almost all derived from hydrocarbons. It's literally everywhere around us... and beneath us.
Q: What does unused oil become (it can't stay a liquid forever right?) when not drilled from the Earth and does it play some important function in the Earth's geological process?
A: There are actually three possible answers to this question, and I'll attempt to address each one:
1. Oil still sequestered in the ground:
Hydrocarbons still in the ground are likely at some sort of equilibrium. After burial of the carbon-rich components (mostly ancient forests and swamps but yes, some dinosaurs also), the carbonaceous material will "mature" with heat and pressure into several final forms: coal, oil, gas. If these cannot escape to the atmosphere (there is some sort of seal, like a salt dome or impermeable sedimentary layer) they tend to stay where they are. If oxygen can get into the reservoirs where the hydrocarbons are lurking, it could lead to further evolution or change of those hydrocarbons, generally an increase in viscosity. Likewise, if the volatile components of crude oil can somehow escape their entombment, what remains becomes heavy crude, tar sands, or coal.
2. Oil that has been taken out of the ground:
Fresh crude, exposed to water and atmosphere, tends to oxidize and self-convert (bio-degrade) to a more sludge-like material. In other words, liquid oil tends to turn thicker or even solid with time and exposure to oxygen and bacteria. When I was a child, my working single mom was so poor that she couldn't afford to change the oil in her car for six years - until the engine seized. The oil pan and engine were full of solid and tar-like hydrocarbons that had to be scraped out mechanically.
There are natural seeps of hydrocarbons in the Gulf of Mexico (that's what clued geologists to start drilling there in the first place). These seeps tend to have evolved benthic communities form around them. This begins with bio-degradation via bacteria. In other words, the sea-life close to a natural seep is different from what you might encounter some distance away.
Keep in mind that there are MANY different kinds of crude oil (API Gravity >10 will float, and API gravity <10 will sink in water, for instance), and they all have different high-viscosity (long-carbon-chain) and volatile (low-carbon-chain) contents, plus assorted poly-aromatic hydrocarbons (PAH's). That API gravity differential leads to an initial separation of the crude oil, if it somehow gets away and flows into water: some of it sinks, some floats, some drifts along in the current. The multi-vis you put in your car has a limited range of carbon-chain molecules compared to the stuff that comes out of the well-head. There are many different exposure environments also, so the speed and degree of change can vary wildly. API > 10 oil from the 2010 Deepwater Horizon well blow-out accumulating at the Louisiana coastline evolves differently than denser oil accumulating at 6,000-meter, near-freezing depths in the deep ocean. Temperature also has a lot to do with how the oil changes with time: higher temperature encourages faster bacterial activity (bio-degradation). There is some evidence that natural seeps on the floor of the Gulf of Mexico have led to different benthic communities based upon the oil and bacterial by-products.
3. Oils that are already used and need to be disposed of:
There are different ways to recycle oil products, but these are as varied as the people doing it. The clean-up of an oil-spill in the Kalamazoo River in 2010 is now estimated to be in the $1.2 billion range. Recycling and clean-up in rivers, sounds, and estuaries may include dredge-and-cap efforts, and may involve storage-in-place, off-site storage, and possibly re-refining or even combustion. A friend collects used cooking oil from restaurants and recycles it; his old Volvo has a sticker on the back that reads "Bio Fuel". The reserves of heavy crude and tar sands in the western hemisphere (mainly Venezuela and Canada) were once estimated to be sufficient to power the industrial world for centuries - if they could be extracted economically. They must to be converted from the solid (or high-viscosity) form first, of course, and this involves vast amounts of heat and water that cannot be used for much of anything else subsequently.