The ticking issue is already open as #8768 (especially #8768 (comment)), but the pdf/evince issue is new. I can't repro using okular (also using poppler as rendering backend) or mupdf or zathura(mupdf) or acroread, so I think it's a bug in evince, but heh.
(Also pdfs generated by mplcairo do not appear to expose this issue.)

Yes, that's true, I couldn't reproduce the PDF displaying issue anywhere else beside Evince.

Thanks for linking #8768. I was wondering no else found this bug before me :-)

Relabeled the issue to focus on the evince problem as the ticker problem is already tracked elsewhere.
Can you report on Evince's bug tracker as well and see whether anything comes out of that?

Maybe for very small ranges change locator:

import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
import numpy as np

arr = np.array([3765, 3792], dtype=np.float32)
plt.clf()
plt.bar(range(len(arr)), arr)
plt.yscale('log')
base_exponents = np.unique(np.floor(np.log10(plt.ylim())))
msb_digits = np.floor(plt.ylim() / (10**np.floor(np.log10(plt.ylim())))).astype(np.int)
if (base_exponents.size == 1 or np.abs(np.diff(base_exponents))[0] < 1) and np.count_nonzero(np.bincount(msb_digits)) < 3:
    ax = plt.gca()
    ax.yaxis.set_major_locator(locator=ticker.NullLocator())
    ax.yaxis.set_minor_locator(locator=ticker.AutoLocator())

This at least puts some numbers on the y-axis. But this doesn't solve the Evince issue.
The output PDF is now missing a part on the left side.

plt.tight_layout()

or

plt.savefig("test.pdf", bbox_inches="tight")

fixes the displaying of the tick labels.

Closing as I think the remaining issue belongs to evince (https://gitlab.gnome.org/GNOME/evince/issues/1056) and I don't think there's anything else to do on Matplotlib's side.