The tip about freezing oil paints to preserve them between painting sessions usually suggested as placing your entire palette in the freezer, is based on the fact that oil freezes are a very low temperature. (It's far lower than water.) Given the temperature a domestic freezer is usually set at, leftover oil paint put into it is unlikely to freeze because it doesn't get cold enough.
We put the question about freezing oil paint to Anne Marie Helmenstine Ph.D. in Chemistry, who said, "The freezing point of linseed oil (the predominant oil in oil paints) is -20°C (-4°F). Most people set their freezers at 0°F, so oil paint will not freeze in most home freezers.
"Oil paints work better when applied at cold or even freezing temperatures, but the paintings themselves are embrittled at low temperatures, especially if the humidity is low. It's fine to store your oils in the freezer if you're taking a break. The cold temperature will slow the rate of oxidation and evaporation, preserving the paint. But once you start a canvas, it's better for the painting to maintain it at a room temperature that's cool but non-freezing, otherwise, the painting may become brittle."
An article in issue no 12 of Golden's "Just Paint" by a "scientist who specializes in coatings" says this about freezing oil paints: "Oils also become more brittle when cold, but embrittlement occurs at temperatures below freezing. ...significant drops in temperature from 23°C down to below freezing at very low humidity can create stresses in a fairly young 13-year-old oil paint film that will exceed its breaking point."1
The tip about submerging leftover oil paint in water to preserve it is a centuries-old one. In a book on color published by the National Gallery in London, David Bomford (Associate Director for Collections at the J Paul Gerry museum in Los Angeles) and Ashok Roy (director of Scientific Research at the National Gallery in London) writes, "...by early Renaissance times professional supplies of pigments existed ... Prepared oil paints were kept in the studio under water to prevent them from drying out."2
Of course, traditional use doesn't always match modern scientific knowledge. Asked about submerging oil paint under water, Anne Marie said: "Exposing an oil painting to water or high humidity is damaging because it can disrupt the cross-linking of polymers, leading to a loss of adhesion (which is naturally poor because linseed oil is not a strong adhesive).
"I would not recommend storing paint or a palette under water, since high humidity inhibits polymer cross-linking, weakening the paint. If the paint is sealed, this serves no purpose except perhaps to prevent rapid temperature swings. If the paint is not sealed, the high humidity can be damaging. Exposure to alkaline conditions may also increase the degree of yellowing over that which would naturally occur."
And to quote from "Just Paint" again, "There are also chemical reactions that can break the polymer chains in oils. The most common is a chemical reaction with water. This reaction is usually slow, but it gets much faster if the paint film is exposed to humid air under alkaline conditions. This becomes a problem if the paint is formulated with alkaline pigments or if it is applied over an alkaline surface."1
So although there's anecdotal evidence from oil painters who've put their oil paints under water and not seen any issues, and it's a practice with a long tradition, it isn't sound on a chemical level. But freezing oil paints is fine if you have a large enough freezer.