Plan B and backup plan are the same thing meaning a plan that you execute if the plan A or the primary plan fails. You can have a plan B even before you start working on the plan A. A back burner comes from the metaphore that you put something on the back burner on the stove in the kitchen and while you deal with the pans that are closer to you, you forget about what you have on the back burner and it gets burned. If you place something on a back burner or as a verb back-burn something, it means that you give little importance to it.

Not the same as a back up plan, as Paul V said it’s a question of priority. You can have or put something (or several things) on the back burner if they aren’t your main priority, but you don’t want to drop them altogether. For example - ‘I started learning German during the holidays but I’ve kind of put it on the back burner now I’m back in work’ ‘A major new project has come in, we have to put everything else on the back burner until it’s completed’ ‘I left my car restoration project on the back burner for years. Now I’m retired, I can spend all the time I want on it’.

In cooking, when soup is ready, the soup is usually moved to a back burner and the heat is set to low. Putting a project on the back burner can mean "set the priority to low" (work on the project only if there is nothing else to do) or can mean "put it hold" (do nothing at this time).

The saying would technically be "to put something on the back burner." It means to set somthing aside, just like you would with a pot on a stove. An example would be, "I was going to school, but I had to put that on the back burner after I had my kid."

It means you haven't forgotten it, you haven't stopped working on it, and you are still making progress on it--but very slowly. You are putting most of your effort into something more important. The idea is that something on the back burner--with the burner set to a very low flame--is still cooking. The cooking hasn't stopped, but it is cooking very slowly.