Does AIC require the residuals of the model to be normally distributed? Does AIC require the residuals of the models to be compared to be normally distributed?
 A: No, but the likelihood function used in the AIC formula should match the distribution of the residuals (see point 3. here). If a normal likelihood is used when the residuals are actually non-normal, that will generally invalidate AIC (except perhaps for some special cases).
A: To answer your question, see below that we are using the letter $f$ twice. That's because this framework assumes you're looking at the right class of models. 
(Twice) the Kullback-Leibler divergence between your model's density $f(\mathbf{x} ; \psi)$ and the true model $f(\mathbf{x} ; \theta)$ is
\begin{align*}
d(\psi|\theta) &= 2 E_{\theta} \left[-\ln \frac{f(\mathbf{x} ; \psi) }{f(\mathbf{x} ; \theta) } \right] \\
&= \int -2 \ln \frac{f(\mathbf{x} ; \psi) }{f(\mathbf{x} ; \theta) } f(\mathbf{x};\theta) d\mathbf{x} \\
&= E_{\theta}\left[ -2 \ln f(\mathbf{x} ; \psi)  \right] - E_{\theta}\left[ -2 \ln  f(\mathbf{x};\theta)\right] \\
&= \Delta(\psi|\theta) - \Delta(\theta|\theta).
\end{align*}
We want to pick the model that minimizes the first part (because the second part is free of $\psi$). AIC approximates that part as the sum of two things. The first part is the expectation of negative twice the log likelihood evaluated using the estimated parameters. You do  not have access to this, but you can evaluate negative twice the likelihood with your estimates--so you have an unbiased estimator. The second part is an expectation that is approximated by a function of the dataset and the number of predictors; this is often called a penalty term. This is cool because we can use our data twice: once for estimating the parameters, and again for approximating the out of sample fit.
Personally, I can't guarantee that you don't need Normality. I suspect you don't, but every time I've looked into the details, I have used a Normal distribution to derive the formula.
A: After reading the resource pointed out by @Richard Hardy, my take-away is:
No, AIC does not necessarily assume that the residual distribution is normal. However, it does assume that the model was fit by maximum-likelihood rather than any other method (such as least-squares). If and only if the residuals of a least-squares fit are normal, the LS model is also the ML model, and AIC is meaningful.
