With many great rivers in the world and many more sources of those same rivers with their many, many tributaries feeding them, it is unlikely that the rivers will die outright. Will the great rivers weaken? Some will, others may become stronger, while new great rivers will arise when the climate shifts disrupt the flow of the older rivers. It is the nature of the world to change.
The Amazon, Mississippi, Nile, Thames, Loire and many others are in no apparent danger of going anywhere, though some have their roots deep into diminishing glacial reserves - the primary threat to Asian river systems - it doesn't seem to affect their yearly levels. The Nile already has lost a headwater source due to volcanic mountain formations and the river loses half its waters to evaporation before meeting the Mediterranean Sea but yet still hangs around. Has it weakened? Perhaps, but even if some of the roots disappear entirely, there will still be other sources available.
The main concern of dying rivers due to global warming rests on the rivers with their primary feeds resting solely on melting glacial waters. These rivers like the Ganges, Indus, Yangtze, and some others are the primary river systems that sustain a source of life for much of India and China, which together house 2.4 billion people. Over 35% of the world's population is dependent upon rivers which continued existence is questionable only helps aid the hype about the death of the great rivers. Hype pushed further by the evidence that the glaciers themselves are shrinking at a greater and greater rate.
Still, there have been plenty of times the world has been without any ice, and yet still gave rise to some of the largest land creatures ever seen. It is doubtful that these great animals were restricted to water sources in only the form of lakes, small streams, and swamps. The great rivers of old were reliant on land drainage from rainfall, which increases in a warmer world. The only thing that changes is the location it falls, and the geographical locations it favors. The modern world differs little.
When will the change occur, and how it will are unneeded questions. How to adapt to the change, accept the new rivers, and compensate for the drying rivers is the only thing that matters now. Although the most recent process was one accelerated by human hands, it is still a natural one. It won't stop simply by reversing our current actions or promoting newer technologies. Whether the current great rivers die or live is truly irrelevant, so long as humanity adapts and moves with the change. Fortunately, whatever the time or state of the world, there will always be great rivers.