One industry-sponsored study claims that only 22% of customers had a positive attitude toward social media advertising, and 8% of consumers abandoned an SNS (Social-networking sites) because of what they perceived as excessive advertising.
Schlosser, Shavitt and Kanfer reached the conclusion that users who "actively contribute content" view online advertisements more negatively than users that only consume it.
Another study found that when SNA delivered content or impressions that provided entertainment or informational value, or offered social value, consumers appeared more likely to respond favorably toward the ad stimuli themselves. It also found that "these results suggested that one prerequisite to more successful SNA execution would be for advertisers to create messages that provide some sort of explicit value to [...] users. And by a wide margin, the value that appeared most likely to be highly regarded by SNS users was entertainment".
Activision ran an extensive advertising campaign when releasing Modern Warfare 2, which included (amongst other advertising channels) Twitter, Youtube and Facebook[4,5]. When the game launched it grossed $550M in what Activision Blizzard calls the "biggest entertainment launch ever".
Deviating a bit on the topic, a study by Nielsen Games polled some 534 "gamers" on whether they responded to advertising in games -- 11% said they purchased a brand, 19% said they talked about it and 10% said they recommended the product, 11% said they sought more information. (While no direct comparison rates were offered against other forms of media, 1% of consumers exposed to direct response advertising eventually buy the advertised product.)
Estimates on how much a "fan" or "like" on facebook is worth ranges between 44 cents and $3.60; the issue seem to be that facebook lack the metrics necessary to correctly analyze advertising impacts on-site. Luckily, facebook has agreed to partner with Nielsen Ratings, and Nielsen Ratings are planning to launch a new "Online Campaign Ratings" system that tracks ad performance and audience demographics.
It's clear that facebook advertising does have an impact, however; "Texas Pete Hot Sauce" ran a campaign on facebook which allegedly had a direct, measurable and effective result.
There are obviously issues to deal with when looking at this data: it's only tangentially related to the question, there are no major studies done on advertising impact on SNS, and I am not an expert in the field of advertising or the research associated with it (I do know my statistics though).
The Activision ad campaign is largely anecdotal; given that we can't see any impact data, but I figured that the academical standards on gamedev.SE is slightly lower than peer reviewed journals, and as such it might be of some interest.
As for a "personal" answer (remember folks: anecdotal evidence is not data), I'd say that yes, a portion of users will probably post stuff on facebook, if it's either relevant or amusing (for instance, Doodle Jump posts your high score and checks your friends list to see if they have any high scores you can compare against. I'd go with "Friends, Fans and Followers: Do Ads Work on Social Networks?" as the most relevant study for your question. Unfortunately, they reach the same conclusion I do:
"Anecdotal evidence indicates that social-networking advertising (SNA) can be effective when users accept it, but the perception of excessive commercialization may lead to user abandonment. Empirical support for these propositions, however, is lacking." and then goes on to look at how users attitudes toward SNA (Social Network Advertising) are affected by things like content, structure and social factors.
: AdReAction. “Brands + Consumers + Social Media: What Marketers Should Know About Who’s Getting Social and Why.” Dynamic Logic, January 26, 2010 [URL: http://tinyurl.com/ycxfewn (dynamiclogic.com) ] -- Removed, but accessible at http://tinyurl.com/3mwmnda (Google Docs QuickView)
: Schlosser, A. E., S. Shavitt, and A. Kanfer. “Survey of Internet Users’ Attitudes toward Internet Advertising.” Journal of Interactive Marketing 13, 3 (1999): 34–54.
: Friends, Fans, and Followers: Do Ads Work on Social Networks?. (2011). Journal of Advertising Research, 51(1), 258-275. Retrieved from EBSCOhost.
 Beirne, Mike. 2008. "Study: Gamers Respond to Ads." Brandweek 49, no. 32: 6. Academic Search Elite, EBSCOhost (accessed August 16, 2011).
 Bruner, Jon. 2011. "What's a "Like" Worth?." Forbes 188, no. 2: 28-30. Academic Search Elite, EBSCOhost (accessed August 16, 2011).