There is no evidence that Crystal Healing works, beyond placebo.
Wikipedia introduces crystal healing as a "pseudoscientific alternative medicine technique".
Wikipedia also claims
There is no peer reviewed scientific evidence that crystal healing has any effect.
While it generally references a book I have never read, Crystal Power: The Ultimate Placebo Effect, it doesn't give a direct reference to support the claim there is no peer-reviewed scientific evidence.
It is hard to prove the absence of scientific evidence, but one standard we use on Skeptics.SE is to find an expert in the area, who has done a search, and rely on that. It isn't perfect (Appeal to Authority), but it is the best we can hope for.
In this case, in 2010, an article was written about oncology in particular:
Pleasant feelings or seeming successes of crystal healing can be attributed to the strong placebo effect, or the believers wanting it to be true and seeing only things that back that up: cognitive bias. A scientific proof of any positive effect beyond a placebo effect does not exist. Even though this treatment can be generally regarded as harmless and without toxicity, it should not be recommended to cancer patients. Thereby we will help prevent our patients from wasting hope, time and money in an ineffective treatment, and at worst to postpone the necessary treatment of this life threatening disease, resulting in a worsened prognosis.
So, in the field of cancer treatment in particular, there is no evidence that crystals can heal.
The Skeptics Dictionary (non-peer-reviewed web-site) makes a broader conclusion:
There is no scientific evidence that crystals are conduits of magical energies useful for healing and protection, or for telling the future.