Is it possible that any life on earth could evolve to live off a liquid which is not water? Something like oil or some other natural liquid?
Water can be substituted with liquid Ammonia (NH3) which has many chemical properties which are similar to water and can fulfill most of the roles which are usually fulfilled by water in organic chemistry.
Ammonia is liquid between -80° C and -30° C. So an ammonia-based biosphere would only be possible on worlds much colder than Earth (or with an atmosphere with a much higher pressure).
However, Ammonia has a problem: It is flammable. When there would be an ocean of Ammonia on a planet with an oxygen atmosphere (and don't think you can get rid of oxygen that easily), one spark would ignite it. Burning ammonia becomes nitrogen and water-ice. So most larger ammonia lakes would likely be below an ice layer.
No, water based life could not swap solvents to ammonia or hydrocarbon. It's far to big of a jump to make. The closest that might happen would be that chemicals from water based biology might be part of biogenesis in a different medium. This would require dumping a significant mass of water based biomatter into an environment with large amounts of liquid ammonia or methane.
An organism that doesn't require water is called a xerophile. Endoliths -- http://www.wikiwand.com/en/Endolith are one example. They live off iron, potassium, or sulfur. Pretty sure there are lots of bacteria that don't metabolize water. And we've witnessed bacteria develop entirely new metabollic pathways. Nylonaise for instance, evolved to metabolize nylon, Escherichia coli evolved a metabolic pathway for ethylene glycol: pathway http://www.ncbi.nlm.nih.gov/pubmed/6336729