Does extra carbon in air help ice melt faster? If you increased the carbon (perhaps more specifically carbon dioxide), in a controlled environment around ice, does it increase the rate that the ice melted?
 A: Carbon dioxide (CO$_2$) can cause heating via the green house effect.
Carbon monoxide (CO) can also cause heating, albeit indirectly. It causes a weak green house effect, but also removes Hydroxols from the air - which help reduce the amount of green house gases.
Methane (CH$_4$) is another green house gas that contains Carbon and is very potent.
Carbon can also be airborne in the form of Carbon particulates (or Black Carbon), which also has a green house effect.
More generally, the rate at which ice melts depends on the amount of heat it receives. It can receive heat from radiation or from diffusion from the air. Green house gases increase the heating from radiation and so are only relevant to experiments where visible light is a significant heating factor. As far as I know, carbon-based molecules don't effect the diffusion of heat in the air (unless they're hotter or colder than the air).
A: Assuming that the question is related to global warming, the answer is "yes", increased carbon dioxide in the atmosphere will cause the ambient temperature to increase because it absorbs infrared radiation and re-radiates it in all directions, leading to an increased rate of ice melting due to the so-called greenhouse effect.  However, there are many contributing factors to consider before drawing a conclusion regarding the size of the effect.  Carbon dioxide only absorbs specific infrared frequencies, and at the current atmospheric carbon dioxide concentration, it is absorbing practically all infrared radiation in that frequency range, so additional carbon dioxide in the atmosphere will have a much smaller effect than expected.  In addition, slightly higher atmospheric temperatures may lead to more water vapor in the atmosphere, which is also a greenhouse gas.  And again, the size of the effect is debatable, because more water vapor can lead to more clouds, which both reflect incoming sunlight and reflect outgoing infrared radiation back to earth.  In addition, more water vapor may lead to more snow at higher latitudes, leading to increased albedo, which reflects more sunlight back into space.
The few factors listed above are just the most obvious set of a very large number of contributing factors, whose interactions are still being quantified by researchers.  These interactions include heat transfer between the atmosphere and the oceans, the interaction between various living organisms and ambient carbon dioxide, and no doubt other interactions that haven't been identified yet.  
