The internals parts
pumps are neoprene, which is only moderately oil resistant, Raritan
in nitrile, which is very oil resistant.
The results is that when oiled with unsaturated veggy oils (corn oil, palm oil) the parts
soften, swell, and work better, for a while, until they wear prematurely. Olive oil and canola oil cause less harm. Or, you can do as it says in the manual, which is to grease annually with silicone or synthetic grease (lots). Tested all of these in the lab.
And yes, oil causes increased odor
and solids build up. How serious depends on how much. A dab now and then won't matter, but a daily dose is a very bad idea; use only enough, as many have suggested... until you can get synthetic grease.