Where is the Buddha quoted as saying do not believe anything I say until you can prove it by yourself? If I remember rightly the Buddha is quoted as saying something along the lines of:


  Do not believe anything I say until you can prove it by yourself


In what text(s) of the Buddhist cannon is this quoted?
The quote comes from Kalama Sutra (AN 3.65) and is often taken out of context, hence misunderstood.

People of Kalama found themselves bombarded by tens of spiritual teachers, each claiming authority and expertise in spiritual matters. These teachers' doctrines were rather different from each other, but each was presented as The truth. Each teacher seemed quite certain of himself and was able to articulate the teaching logically.

When Buddha on one of his tours around the country arrived at Kalama and presented his teaching, the citizens honestly told him, that what he posits as The truth, to them looks like yet another teaching. "Is there any way" they asked, "that we can figure out which of these teachings is real?"

And that's when Buddha gave his famous answer, the point of which:

It is by its results that a teaching should be evaluated.


A teaching can be elaborate and logical, with precise definitions. According to some people's preconceptions, these are the marks of a true teaching.
A teaching  could be profound, deep and mysterious. Some people assume, if chasm is deep and they can't see the bottom, there must be something in there.
A teaching could match student's view of the world, e.g. the scientific worldview, or a spiritual worldview, or both. Many people interpret Kalama Sutra this way, that they should not believe a teaching unless it matches "common sense". They don't seem to realize that what they assume as common sense is in fact the very tangle of preconceptions that holds them in Samsara. 
A teaching could go against student's preconceptions and introduce a completely new theory of everything. Some students are very excited about such esoteric teachings and their eyes glaze over teachers that, in their opinion, profanate Dharma by assuming it speaks about our everyday lives.
A teacher can look confident and speak well, or be soft-spoken and funny, like Dalai Lama. Many people find it hard to relate to a teacher who mismatches their archetype of Sage or Wise Old Man.


According to Buddha, all these are secondary factors, that can't be used as identifying markers of Sat-Dharma (True/Eternal Law/Tradition). Instead, it is by the effects it brings out, both in student's psyche as well as in the world, that a teaching should be measured.

Sat-Dharma is famously good in the beginning, good in the middle and good in the end.

Good in the beginning means, even the outermost layer of Dharma, the one seen by non-Buddhists, has good influence on people. Even the laypeople who don't really practice, but are merely guided by basic Dharma principles, benefit from it. They find that Dharma not only happens to match their highest secular moral and wisdom, but that while secular moral is often too flexible, the compass of Dharma never wavers. When followed at large, True Dharma must lead to reduced suffering and increased harmony in daily lives of common people.

Good in the middle means, when someone practices a slightly superficial version of Dharma, without fully understanding it yet, it greatly reduces amount of suffering one generates inside and around. Student learns to watch his mind and recognize its state, learns to stay mindful of the body and notice arising emotions, learns to not let harmful thoughts and emotions control him, learns to let go of attachments and preconceptions. This leads to increased quality of life, as the student can now stay cool through various life challenges.

Good in the end means, one eventually arrives at Liberating Realization, whereby one is no longer dominated by arbitrary formations, but can instead juggle formations freely.

While some kind of "Good in the end" is obviously the goal of all alternative teachings, it is "Good in the middle" and "Good in the beginning" that is a characteristic mark of True Dharma.

So when Gotama said, "do not believe anything I say until you can prove it by yourself" (or however you want to put it), this is what he meant. He did not mean we should reject a teaching unless it matches our preconceptions. He meant we should evaluate a teaching by its effect on our lives. The proof is in the pudding.