The only other case that comes to mind would be when another Church (the Orthodox, e.g.) or ecclesial community (e.g., Episcopalian, Baptist, Reform, Methodists, etc.) requires some form of its own for validity.
To my knowledge only the Orthodox would require a sacred rite for validity. The others do not.
Dignitas connubii, the instruction on processing marriage cases, discusses when “an ecclesiastical judge must decide about the nullity of a marriage of baptized non-Catholics.”
Here, article 4 §1, 2º says: “in regard to the form of celebration of marriage, the Church recognizes any form prescribed or accepted in the Church or ecclesial community to which the parties belonged at the time of the marriage, provided that, if at least one party is a member of a non-Catholic Eastern Church, the marriage was celebrated with a sacred rite.”
So if an Orthodox wed only civilly before a judge or justice of the peace, subsequently divorced, and now wanted to marry a Catholic, a Catholic tribunal might declare it invalid due to lack of sacred rite.