If many people did this, it would quickly become an incredible annoyance. Have some respect for the commons!
I guess in a way that can get your site some publicity, but I gather that's not exactly what you're after.
That said, I imagine I'd be more okay with it if every time I said yes I got a free coffee.
Aside: I wonder how different reactions would be in different countries/cultures.
Also a supposedly common scam is to get personal info such as names and phone numbers in surveys and, later, to use this information to make fake kidnapping calls sound more plausible.
The upshot is that you learn to ignore people who try to get your attention. I don't think it would work too well here.
Still: the method has significant costs. That's a good think in the "preventing overuse" department.
Don't be afraid to approach people and ask for their help! Most people would love to know that someone cares about their opinion. There are huge online communities that run entirely off this principle, like Yelp or Quora. If someone's not interested, they'll tell you.
I'd hate people walking up to me at random inside the coffee bar (UK reserve perhaps).
PS: to US Starbucks' shops run a tab?
I'd be interested to see if there's any research on this.
Not on that specifically but the concept in general is "reciprocity"
http://en.wikipedia.org/wiki/Robert_Cialdini
In any case offering someone something in advance will make them feel obligated in some way and for that matter even put them in a better frame of mind in general.
My feeling is that if you want honest feedback what I've always done is say "tell me what you don't like about this" rather than "what do you think". That way the person is freed to be critical.
1) If someone does this in my favourite coffee shop, they're polite, and they take my first "no" for an answer, I'm completely OK with this.
2) Having done similar things in the past, it's my experience that - with the caveats above - most people are OK with this.
Obviously, some level of social skill, ability to tell if your presence will be seriously unwelcome, and general politeness and diffidence is rather necessary. Don't send the guy who never washes and keeps getting complaints from women at parties. But if you pass that hurdle, you'll be fine - and as someone said below, it's not like there are enough startups out there that this is ever going to become a major problem.
Just put a note on your table, and if people come to you it's fine. If you are afraid of false positives, offer the coffee giftcard after the review.