There is some weak evidence that knowing that a placebo can work even if the patient knows it is a placebo.
The study compared patients who received no treatment (the control) with patients who received sugar pulls and were told they were sugar pills, but that they might experience placebo effects.
Two-group, randomized, controlled three week trial (August 2009-April 2010) conducted at a single academic center, involving 80 primarily female (70%) patients, mean age 47±18 with IBS diagnosed by Rome III criteria and with a score ≥150 on the IBS Symptom Severity Scale (IBS-SSS). Patients were randomized to either open-label placebo pills presented as “placebo pills made of an inert substance, like sugar pills, that have been shown in clinical studies to produce significant improvement in IBS symptoms through mind-body self-healing processes” or no-treatment controls with the same quality of interaction with providers.
Placebos administered without deception may be an effective treatment for IBS. Further research is warranted in IBS, and perhaps other conditions, to elucidate whether physicians can benefit patients using placebos consistent with informed consent.
These were only mild effects and, by the authors own admission, this was only a "proof-of-principle" study.
There was some criticism about the strength of the result - Ed Yong summarised some in Discover Magazine's Not Exactly Rocket Science blog
In particular, it wasn't clear to what level (a) whether patient's misunderstandings of the contents of the pills affected expectations, and (b) whether patients' expectations affected the result.
This second omission makes this a less than ideal answer to the OP's question. Yong reports that a follow-up publication was expected to answer this, but I have been unable to find one. (Did I miss it?)
David Gorski of the Science-Based Medicine blog was also critical - mainly about the hype surrounding this result, but also suggesting that the patients still had an expectation of some success. Again, this undermines this as an answer to the OP's question.
While we are piling on weak evidence, there was an older study with no control group that showed with no control group that neurotics with no control group openly given placebos with no control group improved, with no control group. It was described by Dr Ben Goldacre in his Bad Science column.
Conclusion: I present no strong evidence that skeptics can or can't benefit from a known placebo, but there have been some interesting exploratory studies that suggest that, just perhaps, they can.