How rare is it for a team to face the minimum number of batters without a perfect game?
If we are just considering games in which a team faced "the minimum number of batters (27)," then not very rare at all.
If we are considering all games, then I'm sure it's rare to some extent.
How often has this occurred throughout history, where a team sends only 27 batters to the plate but is not the victim of a perfect game or no-hitter?
This baseball reference search (criteria: plate appearances = 27; innings = 9; on the road - since road teams are guaranteed to bat for 9 innings in 9 inning games) suggests that road teams on this list had 27 plate appearances (thus, 27 batters) in a 9-inning game. Of the 11 games I can see without a subscription, there was a hit in 6 games (taking a perfect game or no-hitter out of consideration in those 6 games).
After sorting the search by hits, at least 20 games had at least 2 hits under the same criteria above, the most recent taking place in 2013.