(Disclaimer: I'm going to mostly focus on Continental reformers here like Luther and Calvin rather than the English Reformation, because it's not really my wheelhouse yet and because it's hard to sift earnest contemporary attitudes from the hyperbolic anti-Catholic, anti-Inquisition sentiment of the Black Legend. I also focus mostly on the Spanish inquisition for my model of who the Inquisition considered generically was burning, because it's the most famous Catholic inquisition as well as the one whose run overlapped the most with the Reformation and Counter-Reformation.)

I feel like I start every single answer I make this way but -- this depends on what you mean by "innocent people". Are heretics innocent people? Which heretics? To many modern eyes (including mine) executing anyone for any of the charges dealt with by the various Catholic inquisitions is unjust -- did Early Modern people of whatever Christian affiliation feel the same way? Did Early Modern people generally object to the thought of capital punishment for heresy? There were dissenting voices, but the idea that the punishment for persistent heresy was death was subscribed to by many Protestant governing bodies, rather than being the sole province of the Catholic Church.

For an Early Modern Protestant or an Early Modern Catholic, the definition of heresy was determined by what views they themselves took to be normal and acceptable. How a typical Lutheran might feel about the Spanish inquisition in Seville burning a "secret Judaizer", somebody who stubbornly kept on practicing their Jewish faith in secret while outwardly pretending to be a Christian convert, was different from how a typical Lutheran might feel about the Spanish inquisition in Seville burning someone for espousing that Luther had the right idea regarding clerical celibacy. (Not that a person expressing such a view would be punished with a death sentence, but the Spanish inquisition in particular became an apparatus for struggling against dissenting elements within Catholic areas, even dissenting elements that were themselves firmly Catholic in origin and certainly elements that were formally allied with Protestantism.) Some of the views that the Spanish Inquisition found to be heretical, like the belief that fornication was not a sin or that Jesus Christ was not the son of God at all, would have been considered heretical by German Lutherans and Swiss Calvinists as well -- other views that the Spanish Inquisition understood that way were classified that way because they smacked of Lutheranism, like the denial of the transubstantiation of the Host.

The vast majority of people punished by Early Modern inquisitions were not killed, but their fates weren't rosy either, and a robust Inquisition presence could have a powerful chilling effect on religious thought. The majority of those who were killed, especially in the Inquisition's earliest and bloodiest days, were often not Protestants but conversos and moriscos, converts from Judaism and Islam respectively who for whatever reason were deemed not to be true converts at all but insidious secret Jews and secret Muslims. There were public criticisms of the harsh treatment of conversos during the Spanish inquisition, and pleas within the Catholic Church as well as within local Catholic communities for more leniency. But how did Luther and/or Lutherans (or Calvin, and/or Calvinists) feel about Inquisition stances toward Jewish converts to Christianity? Luther in particular is not known for enlightened attitudes toward Early Modern Jewishness, but Protestant views on actual contemporary Jewish people rather than Old Testament Jews varied from sharply jaundiced ("they killed Christ, so they're doomed to misery and servitude and death and they deserve it, Christ's blood is on them and on their children after all") to weakly positive ("Protestants are persecuted, Jews are persecuted, so we have that in common, I guess, maybe we can leave them be"). There were some predominantly Protestant areas such as the Netherlands where Jewish people's religious practice was tolerated under the latter premise, but Catholic worship was not -- there were Jewish converts to Protestant faiths as well as to Catholicism, and many Protestants like Luther had hopes that Protestant Christianity would be more effective in persuading these remaining, supposedly reluctant Jews to convert than Catholicism was. So Protestant loyalty to their own religious faction may have overridden their distaste for people of Jewish descent, and Protestant reformers' enthusiasm for (Christianity-flavored) Hebrew language study may have conflicted with some of the Spanish inquisition's hostility toward continued use of the Hebrew language. But there was no real unilateral ally for Jewish people (both practicing Jews and earnest conversos and anybody in between) against either Catholic persecution or Protestant animosity; there was certainly no obvious natural ally for Muslims and Muslim converts in all this, given that they were seen by Christians in general as even worse than Jews.

Protestants naturally objected to the Inquisition's function in persecuting Protestant thought, but it was because they themselves were also Protestants, not necessarily because all Protestant areas were bastions of religious tolerance in comparison with Catholic areas. It would have been more than a little hypocritical for mainstream Early Modern Protestants to express their shock and disgust at the idea of capital punishment for heresy. Luther supported capital punishment for heresy, as did many self-identified Lutherans. Mainline Protestants despised Anabaptists as much as Catholics did and perhaps more, seeing them as radical splitters, and out of enthusiasm for irony took to drowning them as well as burning them. The Calvinist community in Geneva famously burned at the stake Michael Servetus, another radical who had fled Catholic persecution to Geneva only to personally run afoul of John Calvin. Death was a certain consequence of pretty much all Reformation-era religious conflict -- whether that was formal execution as a heretic, or death in the course of a bloody and protracted civil war, or getting your head bashed in during a brief but spirited religiously-motivated street riot. (It would also be hard to overstate how ubiquitous public punishment as well as corporal and/or capital punishment were in Early Modern European life -- they were not the sole provenance of the Catholic Church by a long shot.) In no way was this bloodshed unilaterally condoned, but neither was it unilaterally condemned; you'd be hard-pressed to find any Reformation-era group whose hands were entirely clean. To a cynical observer, Protestant outrage about the excesses of the Catholic inquisitions would have come across like outrage that they were burning the wrong people -- not that they were burning them at all.

Mentioning links to previous questions is not intended to discourage follow-on questions -- more questions, information, and debate is welcome.

/u/cdesmoulins recently posted replies to "Was there anyone tried for heresy, witchcraft etc during the Spanish Inquisition who survived, or did accusation come with a 100% death rate?", https://www.reddit.com/r/AskHistorians/comments/7zcgwj/was_there_anyone_tried_for_heresy_witchcraft_etc/ . They provided a table of all those tried, and of thsoe how many were burned, by the Spanish Inquisition 1540–1700: 1.8% were executed, 826 out of 44,674 trials.

In an earlier reply, in "was the inquisition as cruel as everyone thinks" at https://www.reddit.com/r/AskHistorians/comments/7wzb77/was_the_inquisition_as_cruel_as_everyone_thinks/ , they talked about more general trends.

EDIT: (I tend to think of points after submitting.) Please do read the articles. There is a lot of detail and of nuance. In particular, the first one above starts and ends by pointing out the unpleasant or even hideous things that could happen to you other than burning at the stake.

So I wonder about the premise of any of the Inquisitions (I believe each country or region had their own) burning a bunch of people.

I can't speak to widespread public criticism from laypeople, but the inquisition was debated and sometimes criticized among intellectuals and the clergy. One interesting intellectual figure in the later days of the inquisition is the German Jesuit monk Friedrich Spee, who focused on how witches are to be uncovered.

Spee's Cautio Criminalis was a harsh criticism of the existing witch trial regimes throughout Europe. Spee begins by asking whether witchcraft exists. His answer is a clear and unambiguous "yes". However, in the following pages of his manuscript, he sets forth several major problems, as he sees them, with the way witch trials are performed. He asks innocently whether we ought to assume witches are guilty, before answering "that's a stupid question". He jokes that witchcraft must be more common in Germany and England than in Spain and Italy, since many more people are condemned as witches in Protestant parts of Europe. He also argues that torture is not to be used in securing a confession of witchcraft, because people will lie simply to make the torture end. Nor can they be condemned solely on the basis of a single witness. Rather, the accused is to be offered an opportunity to defend themselves, and only when their guilt is certain (they must otherwise be presumed innocent!) can they be punished. Spee even outlines a nightmare scenario where the inquisition comes to a town on suspicions of witchcraft: they begin by accusing a few people, who, under duress or torture, implicate others as witches or as accomplices, who in turn implicate others, and before long, the entire town is "proven" guilty even though there have never been any witches there.

Spee was one of the first people to refer to the maxim that the accused must be presumed innocent until proven guilty. This idea is not a new one: the notion of a presumption of innocence existed in medieval English law, and on the Continent, it dates back at least as far as the French jurist Jean Lemoine. Lemoine considered God's exchange with Adam in Genesis 3 as a model for jurisprudence: even though God is omniscient and knows Adam is guilty of disobeying his command, he still offers Adam the chance to defend himself. So there was ample existing legal ground on which to be critical of the Inquisition's methods.

Spee himself was unfortunately ahead of his time. He was seen as "soft" on witchcraft for his condemnation of torture and subsequently stripped of his academic positions. However, he had a strong influence on the Italian jurist and criminologist Cesare Beccaria, who is sometimes called the "father of criminal law" and whose writings influenced the Founding Fathers of the US.

For more, see legal scholar Kenneth Pennington's paper on the history of the presumption of innocence (Pennington somewhat downplays the presumption as it existed in English law, but it is still a good read).