I speak a lot from experience in this regard as I am an out asexual college student--at the age where most people are discovering when the whole chastity/abstinence thing is for the birds. This leads to many different disconcerting scenarios and encounters with said 'dangers'. These dangers are the dangers of navigating a sexual society while determinedly non-sexual.
One of the biggest problems I've encountered is the issue of mixed signals. Asexuals aren't always aware that they're giving mixed signals to others. What some people see as conversation is flirtation to others. Once I got called out on this--I had a long conversation with a fellow male student I didn't know very well as we bitched about our chemistry teacher. When the student left the room, another one of my male friends teased me for flirting with him. I dismissed this immediately, but later I thought... oh God, what if that student did think I was flirting with him? It's a common pitfall--you may see a conversation as having no undertone at all, but that's not necessarily the message the other person is getting. I'm sure this is a problem amongst sexual people as well, but the situation becomes much more awkward amongst asexuals (many of whom don't pick up on this sort of thing at all).
Another related issue is convincing others that your orientation is separate from other aspects of your personality and how you act. I will admit to having the capacity of a horribly dirty mind. I don't typically think of others in a sexual fashion, but by God, if there's a good dirty joke to be had I will find it. I also try to dress attractively, not to attract others but because I like to think I have some sort of aesthetic sense. Yes, I dress cheaply, but I will dress at least cleanly and nicely. Oh, and did I mention I have a penchant for dresses and shirts that take advantage of having boobs? I mean, I don't label them, but I know which shirts in my closet are "boob shirts".
But these aspects of my personality and preferences are entirely separate from my sexual orientation. Just because I'm asexual doesn't mean I can't dress nice, and I find that being removed from sexual attraction allows me to make some very refined dirty jokes. Of the asexuals I'm personally acquainted with, only one is actually a bit squeamish of sexually-oriented conversation. On the opposite end of the spectrum is another asexual friend that is completely averse to having sex herself but INCREDIBLY dirty-minded. She grabbed my boob. I do not approve of this action. Oh well--at least she grabs boobs equally among the sexes.
Still, people tend to take dirty-mindedness and clothing as a reflection of personal sexuality. The best solution I've seen is time. There's not a quick-fix for something that a lifetime has drilled into someone, and in a sexualized society these connections get laid on pretty thick. This leads back to the whole visibility issue. The visibility and knowledge of asexuals will lead to these connections getting worn away bit by bit, and hopefully this will lead to a better overall understanding of how sexuality works.
And this subject leads right into my next one, the question of asexual tastes! But that is for next time.