Are you talking about the closed-cell foam insulation with the off-white "elephant skin" texture on the inside?
If so, that's known as Ensolite, and no, most people do not remove it. It provides needed insulation in addition to making a presentable wall surface.
It does sometimes get removed because it was too dirty, smokey, moldy, torn, or etc. (although many times I think it could have been saved, by cleaning). But then people usually look to recover the walls, although it is a project. (Ensolite is still available, although since it does not have the off-white skin anymore, you paint
it like a wall.)
Without insulation the trailer will be a bit noisier and less insulated. Also it will let more light
in through the walls, and you will have condensation issues in certain temperatures.
There were some brands of trailers that came "plain" (very early Bolers, Compacts, etc.), but most people seek to install something now.