We would expect solar output to influence our climate. If the sun goes out, things will likely chill down a wee bit. No surprise.
On the other hand, there's a politically important (re: almost all GOP) group of eccentrics who argue that CO2 isn't really driving global warming; instead the sun is doing it - either directly or through mysterious interactions with the earth's magnetic field. The implication is that there's nothing we can do about global warming, so we shouldn't talk about a carbon tax (or less efficient versions thereof like emissions trading).
RealClimate has an excellent review of the science involved, starting with the honest scientists and them moving quickly downwards ...
...Work on the influence of solar variability (and on its close cousin, the influence of the Earth's magnetic field) tends to fall into one of three categories. There is the Good, in which careful scientists do their objective best to unravel a complex and probably small (but nonetheless important) signal. As examples of work in this category, I would mention Judith Lean's tireless efforts on relating luminosity to sunspot number, the work of Bard and colleagues on developing isotopic solar proxies like 10Be, Shindell's work on response to solar ultraviolet variability, and the work of Foukal et al on factors governing solar irradiance variations. I would also include the recent work by Camp and Tung diagnosing the amplitude of the solar cycle in temperature in the "Good" category; that it is an easy paper for greenhouse skeptics to misquote takes away nothing from the quality of the science. In fact, I'd say most work on climate and solar variability falls into the Good category. That's rather nice. In fact, scientists have long recognized the importance of solar variability as one of the factors governing climate (see the very scholarly review of the subject by Bard and Frank, available here at EPSL or here as pdf) An understanding of solar variability needs to be (and is) taken into account in attribution of climate change of the past century, and in attempts to estimate climate sensitivity from recent climate variations. Further, the Little Ice Age demands an explanation, and solar variability at present provides the only viable possibility. (It's less clear that the Medieval Warm period is a sufficiently coherent phenomenon to require an explanation).
If the sun were significantly contributing to global warming, by the way, that would logically require us to restrict CO2 emissions ever more radically, since that would be the only part of the equation we could influence.
In a similar vein, critiques of climate models (which appear to have more science behind them) increase our uncertainty margins into a range that includes rapidly catastrophic climate transitions -- such as melting Greenland ice within 15 years instead of 100 years. So these critiques of modeling, which I think are interesting, make restriction of CO2 emissions even more urgent.
It's Reason vs. the GOP again, and we need every RealClimate post we can get.