By what authority would Cardinal Burke judge the Pope a heretic?

Whatever he imagines his authority to be, it would be an authority in opposition to Catholic Dogma:

The Sources of Catholic Dogma, Denzinger

1830 [Recourse to the Roman Pontiff as the supreme judge]. And since the Roman Pontiff is at the head of the universal Church by the divine right of apostolic primacy, We teach and declare also that he is the supreme judge of the faithful [cf. n.1500 ], and that in all cases pertaining to ecclesiastical examination recourse can be had to his judgment [cf. n. 466 ]; moreover, that the judgment of the Apostolic See, whose authority is not surpassed, is to be disclaimed by no one, nor is anyone permitted to pass judgment on its judgment [cf. n.330 ff.]. Therefore, they stray from the straight path of truth who affirm that it is permitted to appeal from the judgments of the Roman Pontiffs to an ecumenical Council, as to an authority higher than the Roman Pontiff.

One sure sign of Diabolical Disorientation is when soi disant Trads begin to introduce novelties to deal with noxious and venomous problems.

The problems of Franciscus will not be solved by any men and the novelty of anyone judging a Pope a heretic is an action that is being too rapidly swallowed by those who are supposed to be horrified by any and all novelty but we have seen that in the last score of years or so, the soi disant trads have not only approvingly swallowed a schism but savored its taste and declared it delectable.

Lord have mercy.

One can not judge a Pope a heretic anymore than one can depose him; however, one can call on the Lord and then wait on Him.

It is HIS Catholic Church, not yours.

The desire to call Franciscus a heretic and then to depose him is an infallible sign of failing Faith (not to say ABS has not been there, done that) and recognising that as a grave failure and repudiating  it is to put one's faithful feet back on the narrow path to Salvation.