I finally found a proper solution to this long-standing problem (currently in Matplotlib 3), which does not require multiple calls to contour or rasterizing the figure.
Note that the problem illustrated in the question appears only in saved publication-quality figures formats like PDF, not in lower-quality raster files like PNG.
My solution was inspired by this answer, related to a similar problem with the colorbar. A similar solution turns out to solve the contour plot as well, as follows:
import numpy as np
import matplotlib.pyplot as plt
np.random.seed(123)
x, y = np.random.uniform(size=(100, 2)).T
z = np.exp(-x**2 - y**2)
levels = np.linspace(0, 1, 100)
cnt = plt.tricontourf(x, y, z, levels=levels, cmap="ocean")
# This is the fix for the white lines between contour levels
for c in cnt.collections:
c.set_edgecolor("face")
plt.savefig("test.pdf")
Here below is an example of contours before the fix
And here below is the same figure after the above fix
