Any time the query generator might produce queries that behave differently on different platforms despite the efforts of the query generator's platform abstractions. Differences in regular expressions, collations and sorting, different levels of strictness about aggregates and grouping, use of full-text search or other extension features, use of anything but the most utterly simple functions and operators, etc.
Also, as you noted, any time you run raw SQL.
It's moderately reasonable to run tests on SQLite during iterative development, but you really need to run them on the same DB you're going to deploy on before you push to production. Otherwise you'll get bitten by some query where different engines have different capabilities to prove transitive equality through joins and GROUP BY
or are differently permissive of queries, so a query will work on one then fail on the other.
You should also test against PostgreSQL on a reasonable data set before pushing changes live in order to find obvious performance regressions that'll be an issue in production. It makes little sense to do this on SQLite, where often totally different queries will be fast or slow.
I'm surprised you're seeing the kind of speed difference you report. I'd want to look into why the tests run so much slower on PostgreSQL and what you can do about it, since in production it's clearly not going to have the same kind of performance difference. I wrote a bit about this in optimise PostgreSQL for fast testing.