Can I keep my Instructor visa?

I have an instructor visa that is valid for almost three more years. I think I might be going home for a year or two to pursue a Masters degree or work and then will return to Japan to live again, and I do not want to surrender my visa unless I have to. My concern is NHI payments and taxes being sent to the address on my residence card while I am gone, so is there any way I can go home for 1-2 years and come back and not have a ton of payments to make? And what is the best thing to do in this situation?

Yes; you just need a re-entry permit. If you leave for up to a year, you just get it at the airport when leaving; otherwis you need to apply for it at an immigration office beforehand.

If you leave for an extended period, you normally complete the move-out procedure at your city office as if you were leaving permanently. Then you are no longer registered as a resident, so you don't pay tax, NHI, etc. And when you come back you do the move-in procedures like you did the first time.

Thanks! So, I just need to fill out the application and inform the city office of the day I will be leaving and complete the move out procedures, and they will give me a re-entry permit? If I try to return in 8 months compared to a year and a half there won't be any problems? If I I am gone for three years and my visa expires will I enter Japan as a tourist for 90 days?

No, you got it wrong. Read again. Did I say that the re-entry permit is obtained from the city office? No, it is obtained from immigration.

You asked two questions, which you may think are related but in fact are quite separate.

First is "can I keep my visa", yes if you obtain a re-entry permit from immigration, and how you do that depends on how long you will be out of the country.

Second is "how do I avoid having to pay NHI, etc. while I am away", and the answer is by completing the move-out procedure at your city office so that you are no longer registered as a resident.

Again, those are two separate things.

You do know that the immigration have the right to cancel your residence status if you do not engage@work.

By right you should informed the immigration changes to your resident status activities.
You can keep your status but immigration have the right to cancel it.
Are you quitting your job and going back, returning before your status expired?

You can get special re-entry permit at the airport by ticking the box for special re-entry permit on the disembarkation card. This will allow you to stay out for less than 1 years without effecting your status.

Anything more than one year, you need to go down to immigration and apply for re-entry permit.

http://www.immi-moj.go.jp/newimmiact_1/en/point_3-4.html

As for taxes, insurance etc. as FIRAS said godown to city hall and make the necessary arrangement.