I suppose that eventually this could happen (people think it might have happened on venus, for example), but not for thousands or even millions of years. But well before this, we’d have rising sea levels due to melting of the ice caps.
There would, however, be some local problems of climate change this century where rivers could dry up in some countries, particulary in the hotter regions as the average temperature changes and rainfall patterns change. So we could see quite big effects on some of the world’s population in the next few decades.
As James says it’s possible that some rivers will start to dry up in the near future, but it is very unlikely that the oceans will dry up because of human made climate change. For the oceans to dry up the Earth would have to heat up lots and lots and probably won’t happen until our sun dies out in millions of years time.
As Suze has said, it is possible, but very unlikely. When the sun comes to an end, then the temperature on earth will begin to rise. The sun will start to get bigger as it turns into a red giant. Whether at this point the oceans and rivers would boil and therefore dry up, or that the atmosphere will be destroyed is unclear. But all we know, is that is a good billion years away. As the others have said, with increasing CO2 build up in the atmosphere, the temperature of the earth will rise. This might only be a few degrees, but can cause droughts as rivers dry up. Although, this increase in temperature might also melt the ice caps, increasing the amount of water in the oceans. So unless there was a huge increase in temperature, the oceans should never dry up.