When will the polar ice caps melt and what will happen when they do?

In addition to rising sea levels, melting polar ice caps have a number of other less obvious effects.
For instance, the world’s seawater traverses the oceans through a process known as the Global Ocean Conveyor. The Conveyor works like this: warm water near the Equator travels north along the ocean surface until it cools enough in the higher latitudes to sink to the deep ocean. Differences in water density allow for water to circulate in this way. The density of water is affected by its temperature and salinity. As the ice caps melt, the incoming freshwater will desalinate the seawater at high latitudes. This will cause the water in the Arctic to become less dense and therefore less able to sink. If the Global Ocean Conveyor is disrupted enough it could cause localized climactic changes such as cooling in Europe and North America.  As reported by NASA, “Without the vast heat that these ocean currents deliver--comparable to the power generation of a million nuclear power plants--Europe's average temperature would likely drop 5 to 10°C (9 to 18°F), and parts of eastern North America would be chilled somewhat less.”
An example of a positive feedback loop, melting ice caps will themselves increase heat through surface absorption. Known as albedo, surfaces have differing abilities to reflect sunlight. As the snow-covered ice melts, it will reveal darker land or water beneath. For instance, sea ice is able to reflect 85% of the sunlight that reaches it – a relatively high albedo. Meanwhile, open water reflects only 7% - a relatively low albedo. What isn’t reflected absorbs into and warms the planet.
Another way melting ice caps could warm the planet with a feedback loop is by releasing gases trapped by permafrost. As rising Arctic temperatures thaw the frozen ground, it will release carbon dioxide and methane - two significant greenhouse gases - trapped underneath.
https://scied.ucar.edu/longcontent/melting-arctic-sea-ice-and-ocean-circulation

https://www.carbonbrief.org/five-reasons-why-the-speed-of-arctic-sea-ice-loss-matters

The colossal impact of global warming is particularly evident when we consider that 2016 marked the boreal ice cap reaching its second lowest measured amount of ice ever seen without any direct or unusual cause. To contextualize, the 2016 number for lowest extent was 4.14 million square kilometers of ice, while the 2012 number (which is the first lowest measured) is 3.41 million square kilometers; however, 2012's numbers can be attributed to unusually warm temperatures that year and to the release of warm water from a broken damn in northern Canada.
What this means is that we are spiraling toward a complete melting of ice caps, with 2016 setting the standard for likely continuance of ultra-low "lowest extent" numbers. As the other educator who responded to this question said, we will likely be seeing ice-free summers as soon as the 2040s. 
The impact of the ice caps melting includes the rise of sea levels. The National Resources Defense Council believes that our oceans will be one to four feet higher by 2100, which could cause huge concerns for coastal cities. Three millimeters a year may not sound like a lot, but it adds up fast. Melting ice caps also disrupt the circulation of the world's oceans, which can impact global temperatures, and erode the habitats of arctic-dwelling creatures, like polar bears. In other words, despite the lax attitudes of many media and/or political figures, global warming is one of the most important issues impacting the sustainability of human life on this planet. If we do not get this issue under control, life as we know it will very quickly change. 
https://slate.com/technology/2016/10/arctic-sea-ice-reaches-second-lowest-extent-on-record-in-2016.html

https://www.nrdc.org/stories/are-effects-global-warming-really-bad

I have good news and bad news.The good news is that the Antarctic ice caps aren't going to melt away completely any time soon. They're simply far too thick; they go down for kilometers. The bad news is that the Arctic ice cap is going to melt away quite soon---at current rates of global warming, most scientists believe it will be effectively gone in about 30 years. The other bad news is that the Antarctic ice caps are in fact melting, and they will soon start contributing huge amounts of water to the oceans.As the ice caps melt, the sea level will rise. If all the ice in Antarctica melted (which, let me repeat, will take a very long time), global sea levels would rise some 70 meters---that's almost as tall as the Capitol Building. This wouldn't completely flood the world---most of the land will still be here---but it would flood almost every coastal city in the world, which turns out to be an awful lot of cities. New York, Boston, Los Angeles, San Francisco, London, Amsterdam... they'll all be underwater unless global warming is stopped or huge dikes are built around them (actually Amsterdam already has dikes---but they'll need much bigger ones). Fortunately we have a fair amount of time to adapt; sea levels currently only rise about 3 millimeters per year, so it'll take about a century for them to rise one foot.
https://www.nasa.gov/topics/earth/features/thick-melt.html

https://www.nationalgeographic.com/magazine/2013/09/rising-seas-ice-melt-new-shoreline-maps/