Well it’s certainly not going to crack the earth in half - that’s not how things work. The earth isn’t just held together like a normal physical object - most of what holds it together is it’s own gravity. But could all the nuclear weapons blow it up? A quick back of the envelope calculation gives that one needs about 10^53 joules of energy to merely overcome the earth’s gravitational binding energy (let alone the force needed to actually break things apart). I’m neglecting the effect of the earth’s rotation there, but it should be about the right order of magnitude.
That’s about 2.5 * 10^37 megatons. Even before I look up some more accurate figures, I can tell you right now that this is way out of the league of the current nuclear arsenal. Maximum yield of a nuclear weapon built so far is about 50 Megatons. In 2002 there were about 40,000 nuclear weapons in the world (according to wikipedia). If we assume those were all 50 Megaton weapons (a gross exageration), that gives us a total of 2 million, or 2 * 10^6 megatons. I don’t think we’ll be blowing the earth up any time soon.
That amount of energy could concievably nudge the earth I guess. Another back of the envelope calculation gives that if you could convert all that energy into kinetic energy of the earth, you’d get a whopping great big velocity change of… 0.0028 m/s. We’re probably not going anywhere just yet unless someone is hiding a spindizzy in their back pocket.
The idea that we could ‘destroy the world’ isn’t entirely wrong though. You could do some seriously bad things with those nukes, and probably scorch the surface of the world. It just wouldn’t do all that much damage to the bulk of the planet beneath the surface.
Anyway, if we ignore the (numerous) technical difficulties associated with setting off a lot of nukes in one place (e.g. if you don’t get the timing exact, they’re going to fratricide and you’ll only get a small fraction of them actually going off), you’re still going to have Bad Things happening if you set them all off. There’ll be a big crater of course, and a lot of crap (much of it radioactive) ejected into the atmosphere. Expect massive climate changes, radiation poisoning, and general unpleasantness. Probably wouldn’t be the end of life as we know it though. 'fraid I don’t have a calculation to back this one up.
If it’s all the same with you, I’d rather not test my predictions though.