What would the effects of detonating every single nuclear bomb currently in existence at once in the center of the earth? Would the earth simply explode into bits? Would the heat have some sort of effect on the surface?
I'll ignore the complete impossibility of getting the world's nuclear arsenal to the center of the Earth and the impossibility of exploding them all at once. The total number of nuclear weapons, including those held in reserve and those scheduled for dismantling, is about 15000, with an average explosive power of less than half a megaton of TNT. This is equivalent to about 7×1018 joules of energy.
That's about how much energy humanity consumes every five days. Compared to other destructive events, it's about 1/5000th of the energy released by the 2004 Indian Ocean earthquake, about 1/70000th of the energy released by the Chicxulub impactor, and about 1/30000000000000th of the energy needed to make the Earth explode into bits (the Earth's gravitational binding energy).
To add to David Hammen's answer. Earth is big. I hate to use the words "really really big" cause there are things much bigger, like the sun, but Earth is quite large.
Imagine what would happen if all the Nukes went off 3000 miles from you. You're in LA, the bombs are all set off in NY. The curvature of the Earth would prevent you from seeing much, except just barely maybe the very top spread of the mushroom cloud and some reflection of the glow and redness in the sky and you might experience a gust of wind from the blast and a short while later, some radioactive fallout and you'd begin to have a bad time within a day or a few days, but the blast probably wouldn't kill you (I'm speculating a bit, but 3000 miles is outside the immediate blast radius.
The Tsar bomb, the biggest nuke ever set off, had a blast radius of 22 miles (35 km) and it leveled a village 34 miles (55 km) away. With the inverse square rule, 100 tsar bombs might have a blast radius 10 times that, or 220 miles and would still knock down buildings further than that. But once you get a couple thousand miles away, even if every nuke in the world is set off, you'd survive the blast, though you'd might feel the pressure wave (wind) generated by it.
Now imagine instead of air separating you and all those Nukes, there's 3,000 miles of rock and metal. You'd barely notice it. Odds are, you wouldn't notice it at all. There's simply not enough heat, and not enough explosive expansion to affect the Earth's core. There would be a pressure wave for sure, but by the time it reached the surface, my guess is you wouldn't even feel it.
There's two kinds of pressure waves that are generated by Earthquakes. S waves and P waves. If you set off nukes inside the Earth's core the pressure waves would primarily be P waves, that would spread out evenly towards the surface.
Using David Hammen's 7 x 1018 joules of energy and Earth's surface area of 5.1 x 1014 square meters. That's about 1400 joules per square meter, which sounds like a lot, but considering that it would be spread out as it travels through the Earth, and compared to local Earthquake energy, that probably wouldn't be much more than a very weak somewhere in the 1-2 on the Richter scale (very bad guess which I'll try to improve on), but based on how much energy a large earthquake has, and if the energy is divided 50% into P wave and 50% into S wave (another bad estimate), 1400 joules per square meter, after having traveled through 3,000 miles of earth would have to be very low on the Richter scale. I'm pretty confident that no person would feel it, but as noted by Jamesqf, it might be detected by sensitive equipment.
The added heat to the core would be even more negligible and I imagine that would have no measurable effect at all.
If someone wants to improve on this answer from a P-wave and earthquake energy perspective, I can delete mine. The bottom line is nothing of consequence would happen.