Is it feasible to envision a biology where oil is the main "solvent" (not sure if this word is chemically accurate). Whereas humans are 80% water, these guys would be 80% "canola oil" or something similar. Would this work? I know that humans and all animals produce small amounts of oil within their bodies already, and some plants are quite efficient at doing so. I don't know if they might need to supplement their oil by direct intake, analogous to our own need to consume water, or if they are self sustaining in that regard, or if they can somehow convert water into oil? (is that what plants do?)
There are a lot of things that would have to change, and lots of ways to handle the problems that come up in making those changes, and the exact set of chosen solutions can make a big difference.
They probably don't breathe oxygen, being either purely fermentative or hydrogen-breathing instead, but there are ways around that.
Is the intracellular fluid the same as the extracellular fluid? Terrestrial cells can keep around oil or fat filled vesicles; do your creatures keep around water-filled (or other polar-solvent-filled) vesicles? And if so, do they use them as solvents for chemistry?
If all chemistry is done in the non-polar organic solvent, then you have to deal with the fact that said solvent is aprotic. That's nice in some ways, because it means that a lot of stuff is far more stable than it is in water, which attacks a lot of biomolecules by hydrolysis and disrupts internal hydrogen bonds, which we water-based creatures have to expend energy to deal with. So, your creatures will have a lower base metabolic load than we do. But, on the other hand, a lot of energy chemistry based on shuttling protons around won't work--unless you have specialized vesicles of protic solvents just for that purpose. Thus, your creatures will probably have to rely on single-molecule electron delocalization and conduction to move energy around, rather than solvated charge gradients.
Additionally, while there is a lot of biochemistry that can be done in non-polar organic solvents, polarity turns out to be really important to the function of some basic structures of our biochemistry. E.g., the repeating charges along DNA backbones don't just make it soluble in water, but the self-repulsion between paired strands helps maintain the doubled-up strand structure, rather than allowing it to tangle up and self-interact like RNA does. Meanwhile, the repeating dipoles in polypeptides are critical in getting them to fold up and self-interact. There is a DNA analog (Peptide Nucleic Acid, or PNA) which is soluble in oil and lacks the phosphate charges (and binds much more tightly as a result)... so maybe your creatures could use that, or something similar, but it's not actually known how well that would work on its own (artificial PNA is used in biotechnology to selectively bind to complementary DNA strands; as far as I know, no one has tried to get it to replicate independently in organic solvent).
Proteins frequently have non-polar sites that are shielded from water, so inverse proteins that have non-polar sites exposed to allow them to be solvated by oil with shielded polar sites for specific reactions might work, but they might just use something completely different. (Some vaguely-specific form of complex polylipid is a popular old proposal for aliens using non-polar solvents, but nobody knows exactly if or how such things would work.)
Cell membranes could still work in basically the same way, just inside-out. But if there is a mixed-solvent system, cells may employ simple single-layer micelles instead.
Plants produce oils, ultimately, from water and carbon dioxide, just like they produce sugars from water and carbon dioxide. The sort of world that would be conducive to the evolution of oil-based life, however, is likely to make that unnecessary. Depending on what the homeworld is like and why your creatures evolved this way, they might consume oil like we consume water, or they might manufacture it (e.g., from some combination of water, methane, and CO2).
Given that some of the fundamental features of cellular life would no-longer function correctly we can't directly infer much about oil based life based just on that fact alone. We'd first need to make substantial wild guesses about how oil based life functions before we could infer anything else.
There are lots of biological functions that leverage the difference between polar and non-polar chemicals. One of the more fundamental of these is the lipid bilayer. If instead of being suspended in an polar fluid, like water, a lipid bilayer was suspended in a non-polar fluid like canola oil they would no longer function correctly.