When I first started taking riding lessons a little over a year ago, I went to a barn where I could volunteer and take lessons. Without going on a rant, I was unfortunately being used, simply because I was so willing to do any crappy job for the sake of being around the horses and learning. My lessons were put on the back burner. I guess they thought I was just SO willing to work that they would get to me whenever. Not to mention the horses were not beginner safe and some of the more "experienced" volunteers and workers treated me like crap.
In my case, I left without warning. I may have hurt some feelings (the owner called and asked where I had gone) but it turned out to be a good choice on my part... After I moved barns and met good horse friends, many told me they had bad experiences with the owner, his horses, and his barn.
You have no obligation to let this instructor in on your business. However, personally if I thought the instuctor was a honest, respectable person who I had built a client/business-type relationship with, I would give them some hint or tell them that you feel you should move on. If you feel they will guilt trip you, give them another excuse.
If the instructor is blowing you off, just runs a poor business, and you have no relationship with them, I wouldn't feel so bad about leaving cold turkey.