Why didn't the Big Bang produce heavier elements? Shortly after the Big Bang, temperatures cooled from the Planck temperature. Once temperatures lowered to 116 gigakelvins, nucleosynthesis took place and helium, lithium and trace amounts of other elements were created.

However, if the temperatures were so high shortly after the Big Bang, why weren't much heavier elements produced? 116 gigakelvins is obviously far above the temperature required for elements like carbon and oxygen to fuse. In addition, shouldn't most of the protons at those temperatures have fused, leaving the Universe with mostly heavier elements?
I think that your thought process is flawed in that you assume that by drastically increasing the temperature you are guaranteed to get heavy elements. As odd as this may sound, this isn't the case (especially during the Big Bang Nucleosynthesis (BBN)) for a few reasons. In fact, if you took a hydrogen-only star and made it go supernova, you wouldn't get heavy elements like you see in current stars going supernova.

BBN Timescale

One major point to consider is that the BBN era is calculated to be only ~20 min long. That isn't really much time to form elements. Sure, supernovae happen in an instantaneous flash, but there are other things going on there, which I'll get to in a second. The main point here is that fusing takes time and 20 min isn't that much time to form heavy elements.

Deuterium

To get heavy elements, you need to build up to them. You can't just smash together 50 protons and 50 neutrons and get tin. So the first step is to smash together a proton and neutron to get deuterium, but here you already run into a problem known as the deuterium bottleneck. As it turns out, the huge temperatures actually (and somewhat counterintuitively) impede the creation of deuterium. This is mainly because the deuteron will end up having so much energy that it will be able to overcome the binding energy (and deuterium has pretty low binding energy being that its only two nucleons) and will likely break apart again. Of course, given the density and temperature you can still get a good amount of deuterium simply by force of will, but not as much and not at the rate you'd expect otherwise. Another point that makes deuterium form less frequently that you'd naïvely expect is that the proton to neutron ratio before BBN was about 7:1 due to the proton being more favorable to be created since it has a slightly lower mass. So 6 out of 7 protons didn't have a corresponding neutron to combine with and had to wait for deuterium to form first before it could combine with anything.

Tritium, Helium, Lithium, Oh My!

Deuterium is then the catalyst for forming all the next stages of particles in your soup. From here you can throw them together with various other things to get $^3\mathrm{He}$, $^3\mathrm{H}$, and $^4\mathrm{He}$. Once you've got a good amount of deuterium, tritium, and helium isotopes floating around, you can start making lithium and if you're lucky a bit of beryllium.

To Boron and Beyond

But now, once again you run into a bottleneck, and one more severe than the deuterium bottleneck. You can't easily jump to heavier elements with what you have on hand. The next fusion chain, and the way stars do it, is the triple-alpha process which helps to form carbon but to perform this chain and build up enough carbon you need a lot of time. And we only have 20 minutes! There just isn't time to form the carbon we need to progress along the fusion cycle. As I hinted at the beginning, pure hydrogen stars also wouldn't produce heavy elements upon supernova for this reason. They're able to produce heavy elements now because they've had billions of years before their SN event to build up a base amount of carbon, nitrogen, oxygen, etc. that can aid in the heavy element fusion processes.

So you don't have the time to follow the triple alpha process and make carbon — what about other processes? Surely the temperatures are high enough that you can do different fusion methods not seen in stars. Well... no. You can't even smash together lots of $\mathrm{He}$ or $\mathrm{Li}$ to get really heavy elements because of the fact that heavy nuclei are only stable if they have way more neutrons than protons. And we already said there was a big neutron deficiency early on so the chance you have enough neutrons hanging around to smash together to get, say $^{112}\mathrm{Sn}$ (that's tin with 62 neutrons), is pretty small. What's more, you can't even try to either skip carbon by making something slightly heavier or form something intermediary between lithium and carbon. Again, this is because of stability issues. So with no other options, you have to shoot for carbon after lithium, and as stated above, you just don't have time for that.

TL;DR

Overall, BBN is limited to getting only to lithium because of limited time, proton to neutron abundance ratios, and fusion bottlenecks that slow things down. All these come together to produce ~75% $^1\mathrm{H}$, ~25% $^4\mathrm{He}$, ~0.01% $^2\mathrm{H}$ and $^3\mathrm{He}$, and trace amounts of $\mathrm{Li}$.