There are generally three types of polar ice:
Sea ice is usually 1-2 meters thick; shelf ice is 100-200 meters thick; sheet ice is one to several kilometers thick.
The poles differ significantly. It's often pointed out that the Arctic is an ocean surrounded by land and the Antarctic is land surrounded by ocean. The North Pole is occupied by sea ice, about half of which melts every summer and reforms every winter.
At the other extreme are the ice "caps," more or less the ice sheets in Greenland and Antarctica that extrude ice in the form of glaciers and ice shelves that continuously flow into the ocean, breaking apart and melting.
To take just Greenland: Greenland has had some degree of glaciation for ~38 million years, but lost much or almost all of its ice during a warming period about 400,000 years ago, suggesting that the current ice sheet was created in that time.
The evidence suggests nearly total ice-sheet loss may result from warming of more than a few degrees above mean 20th century values, but this threshold is poorly defined (perhaps as little as 2 °C or more than 7 °C).]5
(That melt ice is equivalent to about 7.3 meters of global sea level rise.)
So, yes, the ice sheets can melt away; how quickly they can melt will be driven by the amount and pace of CO2 emissions; the extent of that melting will be affected by the duration of that CO2 in the atmosphere and the impact of potential feedbacks.
The formation of an ice sheet -- glaciation -- can occur relatively rapidly in the correct conditions, according to this IPCC Assessment report on paleoclimate, which notes glaciation occurring in some regions of the world within a period of centuries.
In the short term, the most obvious impact we're seeing from global warming is the steady decline in the extent of Arctic Sea ice, at about 2.6 per decade since the satellite record began in 1979. Research suggests that this ice can come back, eventually:
The central finding of this study is that sea ice loss is fully reversible in a state-of-the-art GCM over a range of CO2 concentrations from the 1990s level to nine times higher. We find no evidence for threshold behavior in the summer or winter ice cover in either hemisphere. Thus if tipping points exist for future sea ice retreat in nature, it is for subtle reasons, i.e., through processes that are absent or inadequately represented in this model.
However, this team of researchers suggested an extensive project to prevent a worst-case scenario:
This loss of sea ice represents one of the most severe positive feedbacks in the climate system, as sunlight that would otherwise be reflected by sea ice is absorbed by open ocean. It is unlikely that CO2 levels and mean temperatures can be decreased in time to prevent this loss, so restoring sea ice artificially is an imperative. Here we investigate a means for enhancing Arctic sea ice production by using wind power during the Arctic winter to pump water to the surface, where it will freeze more rapidly.