There was no “constitutional” way to get rid of an emperor. An emperor could lose the support of the pope and/or the German nobility, which happened a couple of times, but they couldn’t really be removed.

As I’m sure you know, starting in the 11th century the popes and emperors were constantly struggling for power. Does the pope have authority over the emperor, and does the emperor have any authority over the pope? The pope had an ambiguous status as the spiritual leader of the church, and as the political leader of the Papal States in central Italy. One of the most significant questions was whether the emperor, and other secular rulers, had the right to appoint bishops within their own territories. Secular rulers typically felt that they did (and so they could appoint bishops who would be more loyal to them than to the far-off pope), and the pope of course disagreed.

The emperor had been much more powerful than the pope and had even appointed the pope and controlled the Papal States in previous centuries, but in the 11th century, starting with Gregory VII, the popes introduced legal and administrative reforms to ensure that the church was in control of appointing and installing its own bishops, and that everyone recognized its authority. The empire was a secular arm of the church, rather than the other way around when the papacy was a spiritual arm of the empire. The culmination of this movement was the Fourth Lateran Council under Pope Innocent III in 1215, probably when the church was at its strongest point. We could probably say the rest of the Middle Ages for the next few centuries was a reaction to this, a constant dispute about who had authority over whom.

The issue of whether or not an emperor can be removed really starts with the death of Emperor Henry VI in 1197. Henry was married to the queen of Sicily, Constance, and their son Frederick, who was only three years old at the time, was set to inherit Sicily as well as the empire, assuming the German nobles elected him. But should one person rule both Germany and Sicily? The Papaal States would be surrounded, so the pope was not in favour of that, and the German nobles weren’t enthusiastic about electing a foreign child. Constance thought she could solve this problem by renouncing Frederick’s rights in Germany and crowning him king of Sicily, but Constance died the next year in 1198. Frederick was now an orphan - and under the protection of Innocent III, because the popes claimed that the kingdom of Sicily was actually a fief of the Papal States (another big issue that would cause lots of problems).

In 1198 the German nobles couldn’t agree on who to elect as the new king of Germany. One faction elected Henry VI’s brother (and Frederick’s uncle) Philip of Swabia, but another faction elected Otto of Brunswick, the nephew of Richard I of England (Otto’s father was married to Richard’s sister). Innocent claimed he had the power to choose one or the other as emperor, and he was clearly in favour of Otto - it would not be in the interest of the pope to have Frederick ruling in Sicily and his uncle Philip ruling in Germany. But Innocent never crowned Otto as emperor and let the conflict drag on until 1208, when Philip was assassinated.

Otto was finally crowned emperor in 1209. But then Otto tried to capture Rome and tried to invade Sicily, so Innocent excommunicated him. The Germans decided to elect Frederick as king of Germany instead. In 1212, Frederick, who was now 18, was installed as king of Germany and king of the Romans, and crowned emperor by the pope. So here, at least, is where your question comes in - how could they elect Frederick when Otto already held all these titles? Could they just decide they didn’t want Otto anymore and elect a new king/emperor? Well there was nothing saying they could, but there was nothing saying they couldn’t either.

Otto of course did not accept this and he continued to act as emperor. From his point of view Frederick was just a rival claimant - just as the emperors sometimes tried to appoint an anti-pope, now the pope had essentially appointed an anti-king. The dispute spilled over into France and England too - Frederick was also supported by the king of France, Philip II, and John of England (Richard’s brother, and therefore another one of Otto’s uncles) of course supported Otto. Otto and John attempted to invade France, but Philip defeated them at the Battle of Bouvines in 1214.

Sometimes historians say Otto “resigned” or “abdicated” but that’s not exactly what happened…he just lost control of the Empire, lost of the support of the pope and the nobles, and lost the actual physical insignia of the Empire, so for all practical purposes Frederick was the emperor. But since there was no precedent for replacing an emperor like this, could he still claim to be the emperor or not? In the end it didn’t really matter, Otto never challenged Frederick, he simply lived out the rest of his life in Brunswick in Saxony, where he died in 1218.

Despite Frederick’s early alliance with the pope he turned out to be the greatest enemy of Innocent III’s successors Honorius III, Gregory IX, Innocent IV. I suppose it’s not necessary to list all the disputes that Frederick had with the popes…we could fill up several more posts with all that and the story is already pretty well known. In brief, Frederick now surrounded the Papal States, just as the Popes had feared, as Emperor and also King of Sicily, and he was extremely ambitious - he also married the crusader queen of Jerusalem and therefore claimed to rule the crusader states as well. He promised to go on crusade, but delayed several times (partly because he was afraid the pope would invade his territory while he was gone), and Gregory IX ended up excommunicating him. Frederick finally did go on crusade and managed to regain Jerusalem, through a treaty with the sultan of Egypt - but everyone thought the point of a crusade was to wage war on Egypt, not conclude a friendly treaty with them, so the church still wasn’t happy.

In 1240 Gregory tried to depose him, but unlike the situation with Otto, this time there was no other legitimate claimant, there would simply be no emperor. Gregory called a council of bishops, but Frederick attacked the fleet carrying the bishops and imprisoned some of them; then Gregory died a few months later and Frederick managed to prevent the election of a new pope for almost 4 years. But Gregory’s eventually successor, Innocent IV, did successfully hold a council, this time in Lyon in France, outside of Frederick’s influence. The acts of the council describe Frederick as

“The beast of blasphemy rises from the sea replete with names, which with the feet of a bear and the mouth of a lion, raging and formed in the rest of its limbs like a panther, opens its mouth in blasphemies against the divine name and does not omit to attack with similar darts His tabernacle and the saints who dwell in the heavens…look carefully at the head, middle and end of that beast Frederick, called emperor, and as far as you find abominations in the words of that man and wickedness, arm sound souls with the shield of truth against the treacheries of that man…” (Rist, The Papacy and Crusading in Europe, 1198-1245)

Clearly there’s no coming back from that…when the Pope thinks you’re the literal Antichrist there’s not much chance for reconciliation. So in 1245 Frederick was officially deposed and the Pope and various other European states tried to install their own new kings of Germany. Frederick was still around and he certainly continued to act as emperor, but how effective could this be if he now had so many opponents? He held on for a few years until he died in 1250. After that, although there were numerous claimants, there simply was no emperor at all until 1312.

So, it was possible to declare that an emperor was no longer emperor, through political manoeuvering by the Pope and the German nobles. But there was no way to enforce it and the deposed emperor could still claim to be emperor until they died.

Sources:

Rebecca Rist, The Papacy and Crusading in Europe, 1198-1245 (Bloomsbury, 2011)

Alfred Haverkamp, Medieval Germany 1056-1273 (Oxford University Press, 1992)

David Abulafia, Frederick II: A Medieval Emperor (Oxford University Press, 1992)