Are there any Protestant beliefs which have their roots in Mormon doctrines?

There are some key areas in which Protestant thought has trended in the direction of Mormon doctrines over the past 200 years.

Of course, Protestants are diverse; none of these trends is universal, and there are counter-currents. And there's near-zero overt, admitted direct borrowing. As you hinted, the primary mode of any influence would be indirect and cultural- perhaps Mormonism influenced changing societal tides that later brought Protestants to shared conclusions. But it's very difficult to dissect cultural shifts to tell what came from that kind of influence and what changes were simply "multiple discovery."

A good reference on this topic appeared years after the question was asked. It's a little book by David Paulsen (RIP) with the tongue-in-cheek title Are Christians Mormon? An earlier version appeared as a journal article and is available for download.

Key areas of 'Mormon-like' shifts in Protestant thought include: (1) spiritual gifts, (2) open canon, (3) the personal nature of God, (4) distinct divine persons, (5) theosis, and (6) the fate of those who never heard the gospel. Three of these trends, (1), (3), and (6), have reshaped much or most of the Protestant landscape, while the others are less widespread.

1. Classical Protestant theology was cessationist. The extraordinary gifts of healing, prophecy, etc. ceased with the Apostles. Mormon belief in spiritual gifts, from Joseph Smith's 1820 vision onwards, was seen as wild and heretical fanaticism. Other continuationists, such as Pentecostals, were likewise seen as fringe or 'cultish' by most Protestants as recently as the 1950s.

   Today, the Charismatic and Pentecostal movements have gained broad acceptance. And many mainstream evangelical ministers today are more open to the possibility of spiritual gifts than their predecessors were a century ago.

2. Classical Protestant theology held that no books outside Luther's list, ancient or modern, could ever qualify as the word of God. The authoritatively closed canon gave a prior reason, before any evaluation of doctrinal content or prophetic provenance, for dismissing the Book of Mormon and Joseph Smith's revelations.

   While Protestants today still have the same 66-book canon, some now say the canon is, in principle, open. There's two main reasons they give:

   Some say treating the fixed canon lists as authoritative is incompatible with sola scriptura. The canon lists came centuries after the Apostles, and they prevailed on the basis of ecclesiastical and political authority, not biblical authority. Some other ancient works could also be theopneustos.

   For others, who believe in the continuation of the gift of prophecy, the canon is in principle open today. As one reverend said, "If this is still the age of the Spirit, there is little argument theologically to say that God has stopped speaking."

3. Classical Protestant theology affirms 'classical theism.' That's a collection of metaphysical ideas about God, developed by philosophers from Aristotle to Anselm &c. Important parts of this are strict and narrowly defined notions of divine aseity, simplicity, immutability, impassibility, and timelessness. Mormon doctrine is incompatible with these narrow definitions, especially impassibility. Rather than Anselm's God, who "is not afflicted with any feeling of compassion or sorrow," Mormon scripture points to a God who weeps over human evil and misery and who rejoices over one soul who repents.

   From the start of the 20th century, some Protestant theologians argued that some of the classical attributes were unbiblical or even incoherent. This led to new theological movements: theistic personalism, free-will theism, process theology, open theism, etc. Even many who still affirm impassibility have redefined it so broadly that their notion would be compatible with Mormonism.

   Outside of scholarly theology, the trend has been even stronger. In 1900 rejecting classical impassibility was a heresy; now it is largely the consensus Protestant position. The influence of Greek and Scholastic metaphysics on Protestant preachers and lay believers is waning. In Pascal's terms, people are more likely to believe in "the God of Abraham, Isaac, and Jacob, and not of the philosophers and scholars."

4. Classical Protestant theology included "Latin trinitarianism." This focuses on the consubstantiality of the Trinity, to the detriment of any doctrine of distinct persons. Mormon scripture affirms that there is one God, but insists that the Father, Son, and Holy Ghost really are fully distinct persons. Mormons do not consider the Nicene and Athanasian creeds authoritative, and thus have no doctrine of consubstantiality. Protestants and Catholics often denounce this as heretical tritheism.

   Today, some Protestant thinkers advocate "social trinitarianism." Each of the divine Persons is a full person, a self which is a distinct center of consciousness, will, and action, yet there is still only one God. Similar beliefs, though without the label, have become more widespread among non-theologian Protestants as well.

   Some thinkers further argue that "Latin trinitarianism" collapses into either modalism or logical incoherence, and that it makes nonsense of the New Testament's account of the relationship between the Father and the Son.

5. Classical Protestant theology emphasized that God is "wholly other" and that there is an infinite gulf between creature and Creator. Mormon doctrines about the saved becoming more like God were seen as a diabolical echo of the serpent's lie "ye shall be as gods."

   But Eastern Orthodox Christians have always seen theosis as a central part of the early Church fathers' doctrine. Since 1950, more Protestants and Catholics have adopted this view. And they've found the doctrine in neglected teachings of Aquinas, Luther, Wesley, &c. Many Protestants now take 2 Peter 1 seriously about becoming "partakers of the divine nature" through Christ. As CS Lewis said in Mere Christianity, "If we let Him... [God] will make the feeblest and filthiest of us into a god or goddess, a dazzling, radiant, immortal creature, pulsating all through with such energy and joy and wisdom and love as we cannot now imagine."

6. Classical Protestant theology held that the unevangelized are predestined for perdition. Mormon scripture declares that all who would have received the gospel if they had received a proper opportunity in life will be saved. Also, in accord with their interpretation of 1 Peter 3:18-20,4:6 and 1 Cor 15:29, Mormons believe the gospel is also proclaimed to the spirits of the dead, and they have a sacrament of baptism for the dead. Traditionally, Protestants found some other interpretation for those scriptures that avoided those conclusions.

   Today, most Protestants are more inclusivist about salvation than was common in 1830. As Billy Graham said, "I used to believe that pagans in far-off countries were lost and were going to hell if they did not have the Gospel of Jesus Christ preached to them. I no longer believe that."

   A few are also more ready to take 1 Peter 3 & 4 seriously as pointing to postmortem evangelization. And while none have adopted a practice of baptism for the dead, some Protestant thinkers now agree that 1 Cor 15 does really refer to such a practice in the primitive Church.

To address the other answer - though Protestants had been diversifying into separate denominations for three centuries, that doesn't mean that their beliefs suddenly ossified just before 1830. No religious tradition is static.

The Short Answer: No. And, AFAIK, there are no Mormon teachings at all that have spread to other traditions. Here's why.

Ecclesiologists who study the history of the church sometimes distinguish between four primary "branches" of christendom: the Roman Catholic church, the Eastern Orthodox church, the Protestant church, and the "bible cults." Protestant is an umbrella term that refers to theological traditions that are rooted in the Protestant Reformation of the sixteenth century. Protestantism is known for its theological diversity, and a quick glance at a World Almanac shows dozens of Protestant denominations worldwide that each report at least 5,000 houses of worship, extending into the millions for larger denominations.

This diversity of theological traditions is due, in part, to the fact that -- unlike the Roman Catholic and Eastern Orthodox churches -- Protestantism lacks a single, unifying authoritative leader or leaders that can define for adherents what is and isn't considered acceptable doctrine. In fact, the Protestant Reformation itself grew out of the rejection of the notion of the authority of clergymen and church history, in favor of consulting the bible itself. The event that is regarded as the "official" beginning of the Reformation was Martin Luther nailing his 95 Theses to the door of the church at Wittenberg, Germany, in the year 1517. The central topic addressed in that document was the official [Roman Catholic] church practice of selling "Indulgences," which Luther believed was an offense against God because, he said, it was not based on sound interpretation of scripture.

So, the word "Protestant" encompasses a wide set of beliefs. In general terms, the Protestant church is differentiated by it's emphasis on the authority of scripture over both church tradition and any extra-biblical teachings of clergymen when seeking to determine what is pleasing to God (see "Sola Scriptura" (1), (2)). There is also a common, though not unanimous adherence to the other "Solas" of the Reformation (see the same references as above). Other than that, however, there are few specifically Protestant points of difference that can be made which will apply to all churches that self-identify as Protestant.

The history of the Mormon church, on the other hand, extends back to 1830 (see "Organization of the Church" sub-heading in the reference). The intervening two hundred-some years between when Protestantism began, and when Mormonism later began, provided plenty of time for the development of many doctrines unique to various Protestant theological traditions. By 1830, many denominations had well established, comprehensively defined sets of beliefs. That is, in many cases there was simply no need to adopt additional doctrine, from Mormonism or elsewhere -- and in many denominations, the process of adopting new doctrine is fairly complex. Finally, there is the widespread contemporary categorization of Mormonism as a "bible cult" by many, though not all denominations. (Note that this C.SE response does not attempt to address the issue of whether or not that categorization is accurate, but simply notes that many people do assert such a categorization).

While my familiarity with Mormon doctrine is at a fairly "introductory" level, and based primarily on a number of conversations with some LDS adherents, my present understanding is that at the basis of Joseph Smith's set of teachings is the notion that all translations of the bible, except for his own, are flawed and incomplete, lacking the additional books endorsed by Smith (the "Book of Mormon"), as well as the modifications made to various books in the Protestant OT & NT, which Smith said were provided to him by one he called "Moroni." (Reference). Since all doctrine established outside of the Mormon church is based on translations other than Smith's, it makes sense that Mormon doctrines remain within those churches that teach Smith's doctrines, and they have not penetrated other traditions.