I can’t find hard evidence for the “lying or crazy” quote, but as OP already mentioned, Feynman said something very similar in content, if perhaps less colorfully phrased:
I think I can safely say that nobody understands quantum mechanics.
Here he is on YouTube saying it.
Here is a discussion of what he meant.
Here is a 2019 New York Times opinion piece with a modern take on the issue by Sean Carroll, a well known physicist.
Because of the similarity between the two quotes, I don’t think it’s of material significance whether he actually made the more colorful statement. It expresses an identical sentiment to what he is shown saying in the video.
As the discussions I linked to above illustrate, the common view among physicists and mathematicians is that while quantum mechanics works amazingly well as a way of predicting the results of experiments, and is a phenomenally successful model of how the physical universe behaves, there is still something fundamentally mysterious and unintuitive about what its predictions really “mean”. That’s likely the sentiment that Feynman was trying to capture with his quote. The point is that he meant something pretty specific.
The reason that this quote has such potential to be used out of context and in misleading ways is that there isn’t an objectively correct definition of what it means to “understand” something. At a practical level, if we can use quantum mechanics to build amazing technological inventions such as lasers, MRI machines, and much more, then we can claim to understand it pretty well. That doesn’t mean we understand everything that we would like to understand, or that it’s some kind of heresy or admission of failure to express frustration about the aspects of the theory we don’t understand, including through pithy, colorful statements.
The issue about the slippery and subjective nature of “understanding” in science is illustrated quite well by another quote from a famous mathematician and physicist, John von Neumann, who once said to a colleague: “Young man, in mathematics you don't understand things. You just get used to them.” See the discussion here.
Sorry for editorializing in the above couple of paragraphs, I realize some people might object that this is off-topic, but I thought it was important to discuss not just the literal question of whether Feynman said something, but also the implied question of whether what he said actually means what some of the people citing his quote seem to think it means.