Are there any known consequences of the right-handedness of the DNA double helix?
The short answer is no, there currently aren't any visible macroscopic asymmetries in some organisms traceable to DNA's handedness. 

The long answer is still no, but I want to clarify that this "no" is more of a "not really". This is because there can be potentially drastic consequences when left-handed DNA is present in a cell in addition to the traditional right-handed DNA.

Let me explain. First, left-handed DNA is officially called Z-DNA. Right-handed DNA, which is the normal DNA configuration, is named B-DNA. There's a third type of DNA, which is also right-handed, that is called A-DNA. A-DNA is essentially super-coiled B-DNA.

Z-DNA isn't that unusual, and it is regularly found in human cells in addition to B-DNA. One study, done in yeast, found that Z-DNA can act as a nucleosome boundary element. Another study, by Kwon and Rich, found that the consequences of Z-DNA weren't as benign. The vaccinia virus requires Z-DNA to bind to a key protein that has anti-apoptosis activity. When Z-DNA does this, apoptosis is effectively blocked.

Furthermore, Z-DNA has been found in patients suffering with Alzheimer's Disease. Suram, et al., found that the hippocampus of severely affected AD patients used Z-DNA instead of B-DNA. This change in the DNA coil has potentially drastic consequences with regards to gene expression.  

Finally, the presence of Z-DNA antibodies were found in patients that suffered from Systemic Lupus Erythematosus (SLE). Lafer, et al., describes how non-SLE patients do not have these antibodies, which suggests that Z-DNA is involved in the clinical manifestation of SLE.

So, with those thoughts in mind, yes, there can be visible asymmetries traceable to DNA handedness. In fact, some tumors depend on genes that have Z-DNA domains (one study by Alan Herbert found that it was about 40% of tumors on their panel). While this isn't really what I think you're asking, it's technically true as tumors do qualify as visible asymmetrical phenotypes traceable to DNA handedness. Diseases associated with Z-DNA could also technically qualify as since they're diagnosable, they're visible in that manner.

Regardless, there is currently no proof that the handedness of our DNA actually causes any real macroscopic phenotypic effect in the developmental stages of life. The reason why left-right asymmetry happens in organisms is due to asymmetric gene expression, not the direction of the DNA coil. This review by Michael Levin talks explicitly about how the left-right borders are established in embryonic development, and there is nothing regarding the direction of DNA's double helix.

Nonetheless, DNA handedness is still a rapidly developing field. I anticipate that more research on these subjects will quickly become available. But as of now, no. Snails shells are not right-handed because of the right-handedness of their DNA.