How would the sketch of a residual plot look for residuals from an exponential distribution with expectation 0? A linear model has been fitted under the usual assumptions, i.e. Y = Xβ + ε, with $ε ∼ N(0,σ^2I)$. How would the sketch of a residual plot look for residuals from an exponential distribution with expectation 0?

Here is a model with $\beta_0 = -8$ and $\beta_1$. As said @BruceET, in order to maintain expectation of $0$, I have to set $\lambda = 1/8$ in the exponential noise. In this case you can view $\epsilon_i \sim \mathcal{E}xp(\lambda) -\beta_0$, thus $\mathbb{E}{\epsilon_i}=0$. What you see on both plots is a random sample from exponential distribution, namely, large concentration at$[-8, 0]$ ( support left boundary up to the expected value) and then random "spikes" upwards.    
set.seed(1)
x = rnorm(1000, 10, 2)

y = - 8 + x + rexp(1000, 1/8)

mod = lm( y ~ x)

par(mfrow=c(1,2))
plot(x, y, pch = 20,
     col      = "darkgrey",
     main     = "Linear Regression",
     cex.main = 0.9)

abline(mod)

plot(fitted(mod),
     residuals(mod),
     pch      = 20, 
     main     = "Residuals vs. Fitted values",
     cex.main = 0.9,
     col      = "darkgrey")
abline(h = 0)


