In his lecture Fallibility of the Popes (November 2004), the late Fr. Gregory Hesse discussed a number of examples throughout Christian history of theological errors made by the Bishops of Rome (not just Honorius!) One such example appears in the (non-dogmatic) section of the bull defining the Immaculate Conception.
In Ineffabilis Deus,
Pius IX gave this warning after the dogmatic definition:
Hence, if anyone
shall dare — which God forbid! — to think otherwise than
as has been defined by us, let him know and understand that he is condemned by
his own judgment; that he has suffered shipwreck in the faith; that he has
separated from the unity of the Church; and that, furthermore, by his own
action he incurs the penalties established by law if he should are to express
in words or writing or by any other outward means the errors he think in his
heart.
As Hesse noted, this is a theological
error and explicitly contradicts session XXIX of the Council of Trent:
[DS 1814] But while
the holy Synod recognizes that those prohibitions by reason of man’s
disobedience are no longer of any use, and considers the grave sins which have
their origin in such clandestine marriage, especially, indeed, the sins of
those who remain in the state of damnation, after abandoning the first wife,
with whom they made a secret contract, while they publicly contract another,
and live with her in continual adultery, since the Church, which does not
judge what is hidden, cannot correct this evil, unless a more efficacious
remedy be applied, therefore by continuing in the footsteps of the holy Lateran
Council [IV] proclaimed under INNOCENT III, it commands that in the future,
before a marriage is contracted, public announcement be made three times on
three consecutive feast days in the Church during the celebration of the
Masses, by the proper pastor of the contracting parties between whom the
marriage is to be contracted; after these publications have been made, if no
legitimate impediment is put in the way, one can proceed with the celebration
of the marriage in the open church, where the parish priest, after the man and
woman have been questioned, and their mutual consent has been ascertained,
shall either say: “I join you together in matrimony, in the name of the Father
and of the Son, and of the Holy Spirit,” or use other words, according to the
accepted rite of each province.
In other words, the Church cannot judge
the thoughts of a person, but only their external actions (e.g., what
they say/do). While not an example of error ex cathedra, Pius IX's comments are an example of a theological error by the pope in a high-level document.
