Categorical claims are often the source of faulty statements. "Don't test with SQLLite [sic] when you use Postgres in Production" by Robellard is a fantastic example. I actually agree with a variant of this statement: "If you need high levels of assurance, don't test with SQLite alone when you use Postgres in production."
Robellard bases his claim on several points, noting that "SQLite has different SQL semantics than Postgres," "SQLite has different bugs than Postgres," and "Postgres has way more features that SQLite." He has a couple more points, but all of these largely amount to a discrepancy between SQLite and Postgres, or between one Postgres version and another, leading to a defect. These points are a genuine concern, but his claim relies on using exactly one database back-end for testing, and exactly one risk profile for various applications.
As a quick diversion, I am not using the common definition of risk which is synonymous with chance. I am using a more stringent definition: "the effect of uncertainty on objectives" as specified in ISO Guide 73:2009. This definition often requires an assessment of both the impact and likelihood of some form of scenario to obtain a fuller picture of an "effect."
If the risk posed by defects caused by an SQLite-Postgres discrepancy is too high, then you'll likely want use Postgres as part of your testing strategy. If the risk posed is sufficiently low, then SQLite alone may be appropriate. These are predicated on the risk posed by defects, and the organisational appetite for risk.
A testing strategy comprising several different testing methodologies can often be thought of as a filter of several layers. Different layers are variously better or worse at surfacing different types of defects. Some are more likely to surface defects within components, and others are better at locating defects in the interactions between components. Other "layers" might be useful for catching other classes of defects. Each layer reduces the likelihood of a defect reaching production, which reduces the risk that defects pose. Each layer also has a cost associated with writing and maintaining that layer.
It's quite common for different layers to be run at different times. For instance, mock-based unit tests might be run very frequently by developers. This provides the developers with very quick feedback on their work. Integration tests backed by an in-memory database might be run prior to committing. These take a little longer to run and so might get run less often, but still catch most problems caused by erroneous component interactions. A continuous integration (CI) server might run integration tests backed by Postgres, and slower UI tests periodically. Finally, penetration tests might be conducted on a yearly or six-monthly basis.
This sort of process aims to allow developers the flexibility to work with confidence by providing quick feedback. However, it also provides heavier-weight checking for the increased levels of assurance required for the risk-averse organisation. An organisation with a greater appetite for risk may remove one or more of those layers, such as in-memory integration tests, to speed development. This saves them money and time but increases their exposure to risk posed by defects.
SQLite is just a tool which may be used as part of one's testing strategy. Declaring "Don't test with SQLLite [sic] when you use Postgres in Production" ignores how it may be usefully applied to reduce risk in a project. In many cases SQLite is entirely appropriate, as the situation simply does not require high levels of assurance. In other cases, it may form part of a more holistic approach along side testing against other database backends, or be removed entirely.
Not every organisation is NASA, and not every project handles secrets of national import. Most failures do not kill people. An honest assessment of the risks would ideally drive the selection of the testing strategy. Often-times this selection will be balanced against other concerns, such as time-to-market and budget. There is no silver bullet. A practical, well-rounded solution is often most appropriate.