One simple immediate consequence may be that you cannot answer a question like "do you have any reason to believe that you may be at risk for XYZ" anymore without potentially having to rely on a result that may or may not have statistical significance -- or which you may have misunderstood. A lot of 23andme reports are in the form of "if you are east asian and have the following genotype for SNP rsXXXXXX, you have a 3% chance of developing condition XYZ". Personal genomics services like 23andme are going out there and saying that SNPs are pretty reliable, but the reality for some conditions and traits may still be fuzzy as more data comes in. There may be as-yet undiscovered SNPs that have a bigger effect on your cancer risk than the ones their v3 chip happens to cover, for example, and just having a 1 or even a dozen SNPs may not be enough for a clear picture. On the other hand, an insurance company might find it risky enough to deny you coverage. Nobody knows how that will shake out.

In the future, if a given insurance provider knows you have purchased a service like this, they also can then figure out which SNPs you know your genotypes for (it would be fairly trivial for any insurance company to find out which 950,000 or so SNPs 23andme tests for), and can then craft a survey form asking about these specific traits.

Many people will likely still try the service anyway because they'd rather know, pro-actively address certain issues, and risk the insurance game, than not know at all.

[EDIT: Apologies for not having read your question all that clearly. Just assume the above applies to "admitting you've used the service in public"]