The Bible is very clear about honoring and respecting political rulers and elders. I think by analogy (if not direct strong implication), these injunctions would apply to the pope as well.
1 Peter 2:17 (RSV) Honor all men. Love the brotherhood. Fear God. Honor the [pagan, anti-Christian, persecuting] emperor.
Exodus 22:28 You shall not revile God, nor curse a ruler of your people. [cited by St. Paul in Acts 23:5 below]
Ecclesiastes 10:20 Even in your thought, do not curse the king, . . .
Titus 3:1-2 Remind them to be submissive to rulers and authorities, to be obedient, to be ready for any honest work,  to speak evil of no one, to avoid quarreling, to be gentle, and to show perfect courtesy toward all men.
Acts 23:1-5 And Paul, looking intently at the council, said, “Brethren, I have lived before God in all good conscience up to this day.”  And the high priest Anani’as commanded those who stood by him to strike him on the mouth.  Then Paul said to him, “God shall strike you, you whitewashed wall! Are you sitting to judge me according to the law, and yet contrary to the law you order me to be struck?”  Those who stood by said, “Would you revile God’s high priest?”  And Paul said, “I did not know, brethren, that he was the high priest; for it is written, `You shall not speak evil of a ruler of your people.’”
Romans 13:1-4, 6 Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God.  Therefore he who resists the authorities resists what God has appointed, and those who resist will incur judgment.  For rulers are not a terror to good conduct, but to bad. Would you have no fear of him who is in authority? Then do what is good, and you will receive his approval,  for he is God’s servant for your good. . . .  . . . the authorities are ministers of God . . .
King David was a mere political ruler, but God made an eternal covenant with him, knowing that he was to commit murder and adultery. David respected Saul when he was king, even though he was persecuting him. This is how even political leaders are accorded respect in the Bible. Popes and King David are similar insofar as they share a trait of being a leader (of a Church or a nation of God’s chosen people: the “ancient Church” so to speak), which the Bible talks about and provides models for.
Yes, David repented. But Nero was the emperor when St. Peter told us to “honor the emperor.” He was slaughtering Christians at the time. He never repented. Peter was martyred during his reign.
Nero was emperor when St. Paul wrote: “Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God” and “he is God’s servant for your good” and “the authorities are ministers of God” (Romans 13:1, 4, 6). He himself was also killed under Nero a few years later.
King Saul never repented and was running around trying to kill David, when David was honoring him as king, and refusing to kill him when he had the chance. David was soulmates with Saul’s son Jonathan, too. How did David react when Saul essentially killed himself after a failed battle, after he had fallen into deep rebellion against God?:
2 Samuel 1:17-19, 23-24 And David lamented with this lamentation over Saul and Jonathan his son, . . .  . . . He said:  “Thy glory, O Israel, is slain upon thy high places! How are the mighty fallen! . . .  “Saul and Jonathan, beloved and lovely! In life and in death they were not divided; they were swifter than eagles, they were stronger than lions.  “Ye daughters of Israel, weep over Saul, . . .”
2 Samuel 2:5-6 David sent messengers to the men of Ja’besh-gil’ead, and said to them, “May you be blessed by the LORD, because you showed this loyalty to Saul your lord, and buried him!  Now may the LORD show steadfast love and faithfulness to you! And I will do good to you because you have done this thing.
David refused to attack or kill Saul, even though he was trying to kill him, because he was “the LORD’s anointed” (1 Sam 24:1-10; esp. 24:6, 10; cf. 26:9, 11, 16, 23; 2 Sam 1:14, 16). All of this for a king who had fallen into apostasy and who rejected God.
It’s argued that God made the eternal covenant with David because he repented of his great sins, and this is true, but it’s also the case that God allowed King Solomon to build His temple (which David didn’t do because he was a man of war). Yet Solomon also fell into serious sin, and seems to have died that way, unrepentant:
1 Kings 11:1-14 (RSV) Now King Solomon loved many foreign women: the daughter of Pharaoh, and Moabite, Ammonite, E’domite, Sido’nian, and Hittite women,  from the nations concerning which the LORD had said to the people of Israel, “You shall not enter into marriage with them, neither shall they with you, for surely they will turn away your heart after their gods”; Solomon clung to these in love.  He had seven hundred wives, princesses, and three hundred concubines; and his wives turned away his heart.  For when Solomon was old his wives turned away his heart after other gods; and his heart was not wholly true to the LORD his God, as was the heart of David his father.  For Solomon went after Ash’toreth the goddess of the Sido’nians, and after Milcom the abomination of the Ammonites.  So Solomon did what was evil in the sight of the LORD, and did not wholly follow the LORD, as David his father had done.  Then Solomon built a high place for Chemosh the abomination of Moab, and for Molech the abomination of the Ammonites, on the mountain east of Jerusalem.  And so he did for all his foreign wives, who burned incense and sacrificed to their gods.  And the LORD was angry with Solomon, because his heart had turned away from the LORD, the God of Israel, who had appeared to him twice,  and had commanded him concerning this thing, that he should not go after other gods; but he did not keep what the LORD commanded.  Therefore the LORD said to Solomon, “Since this has been your mind and you have not kept my covenant and my statutes which I have commanded you, I will surely tear the kingdom from you and will give it to your servant.  Yet for the sake of David your father I will not do it in your days, but I will tear it out of the hand of your son.  However I will not tear away all the kingdom; but I will give one tribe to your son, for the sake of David my servant and for the sake of Jerusalem which I have chosen.”  And the LORD raised up an adversary against Solomon, Hadad the E’domite; he was of the royal house in Edom.
Despite this, we don’t see anywhere (as far as I know) that he should not have been honored as king by the people.
We have no record of the high priest during St. Paul’s trial becoming a Christian or ceasing to oppose Paul. Yet Paul shut up as soon as he was informed who had him struck, saying, “You shall not speak evil of a ruler of your people.”
Now, to be clear: I’m not saying that no one can ever say or do anything about a wicked ruler. The Bible also contains Revelation 13 as well as Romans 13, as I noted recently to a severe critic on another page. I have taken the same view of popes: one can criticize under rare circumstances, with the right attitude and spirit (and the right people doing it). I’ve literally expressed that view for 20 years online (while the reactionaries denied the entire time that I did).
What one cannot do, and pretend to be honoring the pope is lambast, bash, condemn, slander, speak evil against, gossip about, spread mere rumors about a pope day in and day out. That is not “honoring” a pope or ruler, as we are commanded to do, in any way, shape, form, or matter.
We simply don’t find models of pope-bashing behavior in the Bible, even as regards one of the most wicked tyrants in history, Nero, or wicked kings of Israel, such as Saul and Solomon.
(derived from a paper of 11-30-17; some additions on 3-26-18)
Photo credit: Saul Attacking David (1646), by Guercino (1591-1666) [public domain / Wikimedia Commons]