Train your brain to stay awake while reading. You need to let your brain know that you aren't reading as a method to help you fall asleep, you are reading because you want to know what the book says. Each time you go to sleep with a book in your hands, your brain registers that and thinks that's the way it's supposed to be. You need to make yourself think otherwise.
Stop reading immediately and wake yourself up if you feel yourself drifting off while reading. Do a few jumping jacks, drink some coffee or take a cold shower. Doing these things will tell your brain that reading shouldn't make it sleepy, it should energize it. Eventually, your brain will learn its new cues and stop triggering the getting-ready-to-sleep mechanism each time a book is in your hands.
Sit up straight. It is much more difficult to stay awake if you are lounging in your bed or lying on a sofa. Find a stiff chair with a straight back and sit in that or even stand. You don't have to read this way always, but doing it for a while will teach your brain that staying awake while reading is what it's supposed to do.
Read books that are interesting to you and that capture your attention. It is difficult for your brain to shut down for sleep if it is consumed with what you are reading. At least until you get over the habit of falling asleep while reading, read books that really interest you. Think about what you are reading; be an active reader. Reading is an activity and many people forget that. Even if you are sitting quietly with a book in your hands, your brain should be doing mental gymnastics.
Read at the proper time of day. If you read a few pages of a book and then go to sleep, you teach your brain to get ready for sleep whenever you read. Figure out what time you are the most awake and read then. For some people, that's right after they wake up in the morning. Others are most awake in the evenings. If you are sleepy every afternoon after lunch, don't try to read then or you will be likely to fall asleep. Take a run first thing in the morning, grab a cup of coffee and sit down and read a little of your book. You may find you have no urge to fall asleep because your body is wide awake.
Try reading aloud. It may sound silly, but reading aloud triggers a different section of your brain. If you are an audible learner, this may even help you learn better. Read the book aloud to yourself in a mirror, and it will most likely be impossible for you to fall asleep.