Why does the outside of a plastic cooking oil bottle get sticky over time?
Some drying oils, such as linseed oil, react with oxygen in air to polymerize, eventually forming a hard film. When the oil is in the midst of change, i.e. short-chain polymers, it's a sticky mess (don't touch wet paint).

Most cooking oils are unsaturated. Having double bonds makes the oil easier to digest and less likely to deposit in the circulatory plumbing. It also makes the oil more chemically reactive. When in a closed bottle, it reacts fairly slowly with residual air, eventually becoming rancid. Spread over the surface of a bottle, the oil is exposed to more air and oxidizes and polymerizes more rapidly.

An experiment to test this would be to coat two pieces of the plastic used for bottle, usually PETE, one with cooking oil (unsaturated) and one with mineral oil (mostly saturated) and check if the cooking oil becomes sticky, while the mineral oil should not.

One other factor, though, is that oils will cause dust in the air to adhere, so that might also cause some stickiness. Place the test swatches in a container covered with paper towels to keep off dust.