There's really no evidence (one way or the other) to support this. The TVTropes page for Your Mind Makes It Real has a few instances in the "real life" section that generally do not support this. Namely, a lot of times, what happens in the dream is actually a result of something happening in "real life", such as breaking your arm in a dream means you were lying on it in real life, or wetting the bed in real life results in you falling into the ocean or something in a dream.
That said, the mind isn't really understood, and there are some cases of people "developing" symptoms of diseases or conditions that is often a result of hypochondrasis (where someone thinks they have a disease, but they don't). There's also varying amounts of evidence of the success (and failure) of placebos (most evidence points to a mental improvement, which can sometimes help fight the actual disease). There's also cases where people undergoing brain surgery felt someone there (even though it was a side effect of triggering the spacial areas of the brain). 494
There's also the phantom limb sensation, where amputees feel their limbs, even though the limb is clearly gone.
So, could the mind make injuries in the Matrix appear in real life? In a physical sense, no, if someone breaks their arm in the Matrix, it isn't going to magically break in real life. In a mental sense, it really depends on how the Matrix works. If we assume that the Matrix takes over the connections between a person's body in the real world and replaces that feedback sensation with only the Matrix, then it's possible that someone could experience some side effects of injuries in the Matrix in real life. Namely, if someone dies in the Matrix, it could result in their body in real life shutting down, as a result of shock, a heart attack, or simply the mind "giving up" and shutting itself down. Other physical side effects, though, are likely fake, or a result of the real life body jerking around (as <PERSON>'s answer says, <PERSON> probably bit his lip when he fell).
In real life, we don't die in real life when we die in dreams because, on some level, we understand that it's a dream, and not real. In the Matrix, that fact might not hold true, so it's possible that dying in the Matrix would still equal dying in real life. 494