BSG had a pretty good accounting on it's relationship to Mormon Theology...
I'm going to try to do a similar item in relation to the OP. I feel Dragonlance is DEEPLY related to Mormon mythology and history, though it's only ONE sided and comes from Hickman. Weis I think brought in other aspects unrelated to it and probably mellowed it out somewhat.
I'm going to put the Mormon theology stuff in quotes (as best I understand it, I apologize in advance if I get anything wrong, I am writing it trying to be fully respectful of Mormons and the LDS religion, any mistakes are simply that...mistakes on my part), and how DL relates to it below each quote.
The Church of Jesus Christ of Latter-day Saints typically known as the Mormons believes that the true church was on the Earth many millennia ago. However, as those who held the authority from heaven were removed from the Earth, and there were no longer any servants of heaven with the correct authority, that the world fell into apostasy. This happened everywhere, not just in the Nations of the East, but also in the West. They call this the great apostasy. During this time period there was no authority from heaven. There was no revelation and no speaking to or from Heaven. In essence, the heavens were closed to man. In 1820 a young boy named Joseph Smith prayed to know which religion was true. At the time there were various Christian sects in the US and he did not know which one to join He prayed and received a vision in which he said he saw two personages. This created the idea of a Father and a Son as two different individuals or beings. It was not this that started his religion, it was several years AFTERWARDS in which he saw an angel and eventually a set of Gold Plates. From these Gold plates he translated a Book called The Book of Mormon. This led to him asking questions and eventually getting authority returned from Heaven. He (and others who followed him) thus became those who could ask and talk with heaven and the power and authority was returned.
In Dragonlance we have the Cataclysm. In the Cataclysm, if you read the Legends trilogy you find out that those who could actually speak for the Pantheon in Krynn were literally removed. Those who were arrogant and took power on themselves lost the ability or power to call down Clerical spells, but those who actually good priests literally were taken away. This loss of authority created a period where there was no one that could speak or call down spells. In essence, they had lost the authority.
This is equivalent to the Great Apostasy in Mormon theology, but is called the Cataclysm in Dragonlance (that and of course the actual physical disaster, which in some ways could also be seen as a similarity to the Fall of the Western Roman Empire. The similarities in the Legends Trilogy in Istar compared to Rome are pretty prevalent).
The start of the return of Clerical powers is then heralded by a Vision (just like a Vision is the item that kicks off the return of it in Mormon theology). This Vision starts a quest (and there was a similar quest in Mormon Theology, though it didn't have to do with dungeons or dragons or a party, it was more visiting a hill and getting instructions of some sort for several years). The party gets a set of plates which restores the faith and belief of the Dragonlance Pantheon. As some have referred to, there are Various Clerics (Goldmoon, Elistan) to which a more detailed analysis can be applied to.
Thus, you have the story of the lost of a faith (cataclysm/apostasy) the vision, finding the plates or book, and the restoration after a long time of Faith and Authority.
If we go even further, as the reason that it was returned was due to those seeking truth (in Dragonlance, one of the big premises is that the party at the beginning had been seeking the truth or true gods, where as with Joseph Smith it was him asking about which religion to join) you can see even greater parallels of the overarching story of the first book of the Dragonlance Trilogy compared to the Joseph Smith story.
This is actually once again REPEATED in the Mina Trilogy (or War of Lost Souls trilogy) but in a little more convoluted way. Once again the Pantheon of Dragonlance have disappeared, but a messenger comes bringing back the power, authority, and miracles of them. She literally is defeating the Dragons that have come to roost and control Krynn with the loss of the Pantheon, using the Clerical powers to defeat them.
Mormons used to believe in the idea that man could eventually become like God. That man was a spiritual child of God the Father. As Children of God the Father, they were also thus spiritual brothers and sisters to Jesus Christ. As such, they could inherit a reward if they righteous to be joint-heirs with Christ.
This idea is also seen twice in Dragonlance, though respectively it can also be seen as a D&D meme that has been going on for awhile. Raistlin is the first of these where he has aspired to gain the Power of Takhisis. He literally wants to become part of the Dragonlance Pantheon, or to replace it. We find out in Legends that he actually can accomplish this goal, though because he does not have anything to balance this out (another Mormon item that is part of Dragonlance that I'll bring out below), all he does is destroy rather than create.
Mina moreso fulfills this destiny (which is kind of weird as with her trilogy it is by Wies and not Hickman) in that she literally also joins the Pantheon. IN a way it is seen that it is because she is a child of the deities in some weird way,a nd thus it is her right or heritage that she can do so.
Though we see this in D&D occasionally (see Midnight and the Forgotten Realms, Cyric, and others, and especially the Baldur's Gate trilogy) this is directly reflected by what used to be in Mormon theology (though I have not heard of it really being taught or broadcast recently, so this seems like it may have changed).
Mormons have a believe that there must be opposition in all things. Everything has an opposite and can only exist if that opposite also exists. Thus, there must be a balance. Light has darkness, Good has evil, Hot has cold, etc...etc...etc
This is probably the one item I dislike about Dragonlance (as I personally disagree with it, but I won't go into that here) but which you see VERY prevalently in Dragonlance. If there is one Mormon Belief pushed strongly in Dragonlance it is this idea that there has to be balance. The Evil side always wants to squash this, but the good guys almost never want to destroy evil, they just want to ensure there is a balance between good and evil instead.
I don't recall the need of this balance idea really existing in D&D that much (of course, much of this is anecdotal on my part) PRIOR to Dragonlance becoming a thing. It became a much bigger deal afterwards, and even focused on in 2e in the alignment area of how Neutrals could act.
This can be seen as a very LARGE focus between the very origins of WHY the wars of Dragonlance occur and WHY the basis between how the Pantheons act towards each other and towards the inhabitants of Krynn occur.
So, I've outline what I see as some of the MAJOR Mormon influences on Dragonlance above, and I'd say that these influences are pretty major plot and story points for those Dragonlance books written by Hickman and Weis. Hopefully that explains why Dragonlance is seen by many as having a very heavy Mormon influence.
I DO really enjoy Dragonlance, it is probably one of my favorite D&D campaign settings, but my love of Dragonlance really has nothing to do with the influence of Mormonism on it and more simply because I enjoyed the stories overall and the organizations (Knights of Solamnia, Holy Order of the Stars, Wizards of High Sorcery, etc).
