This is one of those situations where we all die, but not necessarily in the way you might think.
An important thing to know here is that water evaporation essentially occurs because individual molecules of water get knocked by other molecules in exactly the right way to push them over '100' degrees. This is still going to happen if the boiling temperature is 200 degrees or 1000 degrees, though it will happen a lot less. The world, however, is really big, and contains a lot of water, so rain would still happen and the water cycle would continue.
Not that it helps much. If global humidity dropped by 50% (I'm just pulling that number out of the air, the actual number relies on a lot of other factors such as weather conditions, temperature and distance from the sea) then the resulting drought would kill a lot of crops we rely on, or just kill a lot of people. It would also cause some major climate havoc.
Water vapour is a greenhouse gas. It's not the most long lived greenhouse gas, but it is a powerful one (due to the large amount of it). Strip that away and the temperature of the planet changes drastically, dropping humanity into a nice ice age. Or possibly just counteracting global warming. It's hard to tell with climate science.
However: That isn't your biggest problem. Your biggest problem is that you've changed one of the fundamental properties of a material we rely on to live. Kii noted in the comments (quite helpfully) that the equation PV=nRT is important here (I know it doesn't quite apply as water isn't an ideal gas, but the same principles are important), but instead of it being important on a global scale it's really, really rather important on a cellular one.
Your body has evolved for water to work a certain way at a certain temperature. It's optimised for it, and in the case of semi-permeable membranes (every cell in the body) relies upon the properties of water remaining the same. Our bodies 'know' that water will move in a certain way at 36 degrees. You've just changed the way water works. Suddenly water that was easy to push through the cell wall isn't. Capillaries that were flowing nicely suddenly aren't. The load on the heart changes drastically. Lymph nodes, brain cells, anything that has water in it ceases to function as it used to, leading to massive organ failure likely with the same set of symptoms as hypothermia.
The human race dies, but not because the rain has stopped.