Can Dickey-Fuller be used if the residuals are non-normal? Can the unit root test of Dickey-Fuller be used even when the residuals are not normally distributed?
 A: Yes, that is not a necessary condition. Recall that all we know about the null distribution of the Dickey-Fuller test is its asymptotic representation (although the literature of course considers many refinements).
As is often the case, we do not need distributional assumptions on the error terms when considering asymptotic distributions thanks to (in this case: functional) central limit theory arguments.
Here is a screenshot from Phillips (Biometrika 1987) stating assumptions on the errors - as you see, these are way broader than requiring normality.

That said, the asymptotic distribution does not have a closed-form solution, so that you need to simulate from the distribution to get critical values (existing infinite series representations are not practical either to generate critical vales). To perform that simulation, you must draw erros from some distribution, and the conventional choice is to simulate normal errors.
But, as Phillips shows, if you were to draw the errors from some other distribution satisfying the above requirements, you would asymptotically get the same distribution. You could replace the line with rnorm(T) with some such distribution in my answer here to verify.
That said, the finite-sample distribution will of course be affected by the error distribution, so that the error distribution will play a role in shorter time series. (Indeed, I did replace rnorm(T) with rt(T, df=8), and differences are still relevant for $T$ as large as 20.000.)
A: Yes, the innovations need not be normal, not at all.
The underlying mathematical fact that gives rise to the asymptotic null distribution of the DF statistic is Functional Central Limit Theorem, or Invariance Principle.
The FCLT, if you'd like, is an infinite dimensional generalization of the CLT. The CLT holds for dependent, non-normal sequences, and similar statement can be made about the FCLT.
(Conversely, FCLT implies CLT, as the finite dimensional distribution of the Brownian motion is normal. So any general condition that gives you a FCLT immediately implies a CLT.)
Functional Central Limit Theorem
Given a sequence of random variables $u_i$, $i = 1, 2, \cdots$. Consider the sequence of random functions $\phi_n$, $n = 1, 2, \cdots$, defined by
$$
\phi_n(t) = \frac{1}{\sqrt{n}}\sum_{i = 1}^{[nt]} u_i, \; t \in [0,1].
$$
Each $\phi_n$ is a stochastic process on $[0,1]$ with sample paths in the Skorohod space $D[0,1]$.
The generic form of FCLT provides sufficient conditions under which $\{ \phi_n \}$ converges weakly on $D[0,1]$ to (a scalar multiple of) the standard Brownian motion $B$.
Sufficient conditions, which are more general than those from  Phillips and Perron (1987) quoted above, were known prior, if not in the time series literature. See, for example, McLiesh (1975):

*

*The strong mixing condition (iv) of Phillips and Perron implies McLiesh's mixingale condition under some conditions.


*Condition (ii) of Phillips and Perron requiring uniform bound on $2 + \epsilon$ moments of $\{ u_i\}$ is relaxed in McLeish to the uniform integrability of $\{ u_i^2 \}$.


*Condition (iii) of Phillips and Perron is actually not quite correct/sufficient as intended.
For a further milestone in the time series literature in this direction, see Elliott, Rothenberg, and Stock (1996), where they apply a Neyman-Pearson-like approach to benchmark the asymptotic power envelope of unit root tests. The normality assumption is long gone by then.
DF Statistic
It follows immediately from the FCLT and the Continuous Mapping Theorem that the DF $\tau$-statistic $\tau$ has the asymptotic distribution
$$
\tau \stackrel{d}{\rightarrow} \frac{\frac12 (B(1)^2 - 1)}{ \sqrt{ \int_0^1 B(t)^2 dt} }.
$$
The 5th-percentile of this distribution is the critical value for DF test with nominal size of 5%.
Simulating $\tau$ with an i.i.d. normal error term and another error term that follows, say, a time series specification would lead to the same distribution as sample size gets large.
Comment
I am going to disagree with @mlofton's comment that

Generally speaking, no one would ever use Phillips result
because simulating analytically from the derived distribution is not
terribly practical. People generally use the DF tables and those have
nothing to do with asymptotic representation. They use the normality
of the error term and allow the practitioner to obtain DF statistics
for sample sizes as low as 20...

It's a major contribution of Phillips to point out that an "assumption free" asymptotic distribution is possible. This is one of the reasons, along with contemporary developments in economic theory, that convinced empirical practitioners (in particular, macro-econometricians) that unit root tests belonged to their everyday toolbox. A statistic (more specifically, a null distribution) that requires normality of the data generating process is not useful at all---e.g. suppose the $t$-statistic is only valid if data is i.i.d. normal. That was the limitation of the early unit root literature.
