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Normally the pitcher who allows a runner to reach base is charged with the run if the runner eventually reaches home, even if the pitcher is replaced before the runner scores. If the run is an earned run, that pitcher's ERA is affected.

For this abbreviated season, MLB is placing a runner on second base at the start of each team's at bat in extra innings. (This was done in the minor leagues for a few seasons prior to this one.) That runner is either the last batter of the previous inning or a substitute for that batter. Either way, the pitcher did not allow the runner to reach base. Thus the question: How does it affect a pitcher's ERA if the automatic runner scores?

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It doesn't. The scored run is considered unearned.

A news article on MLB's own website describes the effect on scoring:

For purposes of calculating earned runs, the runner who begins an inning on second base pursuant to this rule shall be deemed to be a runner who has reached second base because of a fielding error, but no error shall be charged to the opposing team or to any player.

while another of their articles (describing all rule changes for 2020) says

If that runner scores, the pitcher won't be charged with an earned run.

A Washington Post article, likely summarising previous analysis, has the same statement:

Pinch runners are allowed, and pitchers will not be charged with an earned run if the designated runner scores.

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