Except for tests that include non-unicode characters:
tests/test_po.py
tests/test_strformat.py
tests/test_url.py
tests/checker/test_error.py
tests/checker/test_news.py
I believe all these tests require the network, at least they seem to
fail if it's I run them without connecting my computer to the web.
I'm looking at this as part of packaging linkchecker for GNU Guix,
where the package is build and the tests are run in a isolated
environment, intentionally without network access, to avoid issues
with non-reproducible package builds.