While we conceive of punishment of heresy by fire as a ubiquitous medieval phenomenon, it was in fact a social phenomenon with specific historical origins. Heretics were not burned through the early middle ages in Latin Christendom; it took off in the 11th century as dramatically exemplified by the first recorded auto da fe of 1022 at Orleans.

That said, the notion of killing and burning those who 'crossed a line' within the social order was certainly not new to the post-Carolingian Frankish population of the 11th century. Capitularies of Charlemagne make it clear that this was going on in the 8th century, for example in this edict of the 789 CE capitulary for the Saxons:

"If anyone, deceived by the Devil, shall believe, as is customary amongst pagans, that any man or woman is a striga [a flying, cannibalizing woman], and eats men, and shall on that account burn that person to death or eat his or her flesh, or give it to others to eat, he shall be executed."

So, this is evidence of two related issues for the Carolingians:

1. The reign of Charlemagne was concerned for consolidation of basic Christian beliefs as it sought to Christianize newly conquered pagans (since the conquests of Charlemagne's grandfather Charles Martel up to and including his own of the Saxons).

2. Heresy was not being a striga, but believing that such things could exist and killing them; the matter, while theologically founded, was a secular crime as it would remain for several hundred years to come.

For the general population,

The faith was defended through affirming the authority of bishops and archbishops: the clergy were trained to understand the Creed, celebrate the Mass and baptism and use penitentials, and their flocks were instructed by means of explanations of the Trinity, the Lord's Prayer, the Ten Commandments and the Creed, and through the reading of homilies.

This expansion involved explanations of the Christian faith to recent converts from Germanic paganism, who had to learn what comprised 'the cup of life and salvation' which they were offered. At the same time clergy familiar with Carolingian education tried to formulate accounts of the nature of the redemptive power of the church and the meaning of its doctrine and worship, raising the threat of heresy and challenging the Carolingian episcopate to pronounce formulae of orthodoxy. ^1

This is as close as we get to a view of broader-based 'heresy' within the general populace, although never named a heresy. This is typical of the Roman Church's separation between beliefs resident among Christians, and beliefs of non Christians: non-Christians (ie Jews, Muslims) can't be heretics (although some in the late middle ages try to work around that.)

However, coming back to Alcuin and the Council of Frankfurt, heresies were a problem that the Carolingian-era Church confronted within ecclesia itself, similar to the experiences of the church in antiquity: questions of theology and Christology. Let's look at little more closely at the results of the large one. The specific heresy here was Adoptionism found in the newly conquered Spanish marches along the Pyrenees, and came to the attention through the (now lost) pro-Adoptionist writings of the bishop Felix of Urgel (although he was not the only bishop to hold this view - others were in Muslim Iberia).

How was the Pyrenean bishop Felix dealt with? It took years of back and forth letters and tracts, councils, discussion, edicts. Interestingly, we actually have an image of these kind of councils from the Carolingian manuscript, the Utrecht Psalter. Finally,

Pope Leo III condemned Felix at Rome in October of 798, and in 799 at the Council of Aachen Felix recanted after a debate with Alcuin, and was deposed and sent to Lyons, where he died in c. 818. ^2

When Felix was deposed he was subjected to house arrest, likely in a monastery associated with the Lyonnaise bishopric.

The foregoing, in addition to other evidence from the period, tells us that the Carolingians were very much absorbed in determining orthodoxy, but it was not a function nor expression of political power or social unrest. This would change a few centuries later when ecclesiastics of the Roman Church would change tactics and sanction the burning of heretics.

Notes

^1 & ^2 For a thorough grounding in Carolingians, theology and heresy, see David Ganz's essay "Theology and the Organization of Thought" in Rosamond McKitterick (Editor), The New Cambridge Medieval History: Volume 2, c.700-c.900 (Cambridge University Press, 1995)

With respect to the political and social context of the Adoptionist heresy in the empire, see Cullen J. Chandler, "Heresy and Empire: The Role of the Adoptionist Controversy in Charlemagne's Conquest of the Spanish March", The International History Review, Vol. 24, No. 3 (Sep., 2002), pp. 505-527

And for burning heretics, no better place to start than R.I. Moore, The War on Heresy, (Profile Books, 2012).