Is "The body never lies" supported in the suttas?
Obviously, it's not really the "body" that is wiser, it is the non-conceptual mind which constitutes the underwater part of the iceberg the top of which we see as our conceptual mind. The question is, should we trust it?

Generally speaking, in western culture the body with its instincts and emotions is seen as an animal - stupid and violent, while the rational mind is seen as all that is good in the human. However, other cultures may have different basic assumptions. At the risk of overgeneralizing, I think that in Asian cultures the ancient heart-mind (the deep emotional mind - what we westerners usually call "the subconscious") - is generally recognized as a source of wisdom, harmony, and the arts - while the modern rational mind is often seen with suspicion as the source of confusion and conflict.

As for canonical evidence,  all we have are some bread-crumble trails that look cohesive enough to some but not convincing to others, as is often the case with Buddhism: 

Buddha of Pali Canon spoke about "mindfulness of body" (kayagata-sati) and "the signless" (animitta). The contemporary Mahayana tradition holds that these are references to tapping into the non-conceptual mind, which is experienced as (psychosomatic) sensations in the body.

Personally, I was given a significant amount of training and instruction on getting in touch with the above through various psychosomatic practices, and in my experience it all comes together quite cohesively and makes total sense. To me these practices were presented as integral part of the teaching and I certainly see them as important as meditation and study of texts, if not more important. 

As for the textual sources, the closest I can think of is Touching Enlightenment: Finding Realization in the Body by Reginald A. Ray (a student of Chogyam Trungpa Rinpoche and a Ph.D). There is also a 100% western work called Focusing, developed by a psychologist Eugene Gendlin in 1970s, which speaks about same thing from a slightly different angle (thanks to Ronald Cowen for pointing out the Focusing book to me). If you are interested in emotional intelligence and psychosomatic methods, whether Buddhist or not, these two books are recommended.

Now, to address your specific concern: you are right that in an untrained person the signals from the non-conceptual mind can come mixed up with all kinds of phobias and prejudices, so a lot of preliminary work is required to clean that up until it can be fully trusted. That said, I do believe that the more someone is "enlightened" the more he or she can utilize the non-conceptual mind to its full capacity, which exceeds that of conceptual mind.