Short answer
No. A feeling in prayer can be real, but it cannot settle which book is from God (﷾). The Book of Mormon asks its readers to pray about it (Moroni 10:4). It promises the Holy Ghost will show them it is true. Many Latter-day Saints speak of a burning in the bosom (Doctrine and Covenants 9:8). But sincere people of other faiths feel the same about books that contradict it. Two books that contradict each other cannot both be true, however strongly each is felt. The Bible warns that the heart is deceitful, and tells believers to test the spirits. By David Whitmer's account, even Joseph Smith said some revelations are of man, and some of the devil. Islam asks for evidence first (17:36). Then the heart finds rest in the truth (13:28).
Full explanation
Moroni's promise
The Book of Mormon ends with an invitation. Readers should "ask God, the Eternal Father, in the name of Christ, if these things are not true." Those who ask sincerely are promised: "he will manifest the truth of it unto you, by the power of the Holy Ghost."
A revelation of 1829 describes the answer: if it is right, "your bosom shall burn within you; therefore, you shall feel that it is right." Missionaries invite people to read the book and pray in this way.
Readers are asked to "ask God, the Eternal Father, in the name of Christ, if these things are not true". To the sincere asker, "he will manifest the truth of it unto you, by the power of the Holy Ghost."
Oliver Cowdery must "study it out in your mind" and then ask. If it is right, "your bosom shall burn within you; therefore, you shall feel that it is right." If not, he will have "a stupor of thought".
Others feel the same about other books
Sincere people in many faiths pray and feel that their own scripture is true. A Hindu may feel it about the Gita, and a Christian about the Bible alone. A Muslim feels it about the Qur'an.
These books contradict each other about God (﷾), Jesus and salvation. They cannot all be true. So the feeling, however real, cannot be what decides between them. It would give every sincere seeker a different answer.
Readers are asked to "ask God, the Eternal Father, in the name of Christ, if these things are not true". To the sincere asker, "he will manifest the truth of it unto you, by the power of the Holy Ghost."
I prayed, and I feel that it is true.
Do others feel the same about other books?
Yes. Sincere people feel it about books that contradict each other.
Can contradicting books all be true?
No. So the feeling cannot be what decides.
Then what does?
Tests anyone can apply: no contradictions, a text that was kept, a history that holds.
And the feeling?
Peace follows the truth: "in the remembrance of Allah do hearts find satisfaction".
The Bible says: test it
Latter-day Saints accept the Bible. It does not make the heart the judge. "The heart is deceitful above all things" (Jeremiah 17:9). "He that trusteth in his own heart is a fool" (Proverbs 28:26).
It tells believers to check: "Beloved, believe not every spirit, but try the spirits whether they are of God" (1 John 4:1). And it gives a test for a prophet's words: if what he foretells does not come to pass, "the LORD hath not spoken" it (Deuteronomy 18:22).
“The heart is deceitful above all things, and desperately wicked: who can know it?”
“He that trusteth in his own heart is a fool: but whoso walketh wisely, he shall be delivered.”
“Beloved, believe not every spirit, but try the spirits whether they are of God: because many false prophets are gone out into the world.”
“When a prophet speaketh in the name of the Lord, if the thing follow not, nor come to pass, that is the thing which the Lord hath not spoken, but the prophet hath spoken it presumptuously: thou shalt not be afraid of him.”
A revelation that failed
The Joseph Smith Papers, published by the Church, record a revelation of early 1830. It sent four men to Kingston, in Canada, to secure a copyright for the Book of Mormon. They "failed to obtain a copyright."
David Whitmer, one of the book's witnesses, later wrote that they asked Joseph Smith why. By his account, Joseph asked the Lord and received this answer: "Some revelations are of God: some revelations are of man: and some revelations are of the devil." The editors note that British law made such a copyright hard to get. But the lesson stands. If even a revelation of Joseph Smith's could come "of man", a sincere inner sense cannot be enough to tell.
A revelation of early 1830 sent Oliver Cowdery, Hiram Page and two others to Kingston, Upper Canada, to secure a copyright for the Book of Mormon; they "failed to obtain a copyright." David Whitmer later wrote: "We asked Joseph how it was that he had received a revelation from the Lord for some brethren to go to Toronto [Kingston] and sell the copy-right, and the brethren had utterly failed in their undertaking." By Whitmer's account Joseph enquired of the Lord and received the answer: "Some revelations are of God: some revelations are of man: and some revelations are of the devil." The editors note that British copyright law made such a copyright hard to obtain.
Islam asks for evidence first
The Qur'an does not ask anyone to believe on a feeling alone. "Pursue not that of which thou hast no knowledge; for every act of hearing, or of seeing or of (feeling in) the heart will be enquired into" (17:36). It warns that "fancy can be of no avail against truth" (10:36).
Instead, it offers tests anyone can apply. Look for contradictions (4:82). Try to produce a chapter like it (2:23). See whether it has been kept (15:9).
“And pursue not that of which thou hast no knowledge; for every act of hearing, or of seeing or of (feeling in) the heart will be enquired into (on the Day of Reckoning).”
“But most of them follow nothing but fancy: truly fancy can be of no avail against truth. Verily Allah is well aware of all that they do.”
“Do they not consider the Qur'an (with care)? Had it been from other Than Allah, they would surely have found therein Much discrepancy.”
“And if ye are in doubt as to what We have revealed from time to time to Our servant, then produce a Sura like thereunto; and call your witnesses or helpers (If there are any) besides Allah, if your (doubts) are true.”
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
Then the heart finds rest
Islam does not dismiss the heart. The Qur'an says: "without doubt in the remembrance of Allah do hearts find satisfaction" (13:28). Many who come to Islam describe deep peace.
But in Islam the peace follows the truth; it does not replace it. First test the book with your mind. Then pray to God (﷾) directly, who says: "I listen to the prayer of every suppliant when he calleth on Me" (2:186).
“"Those who believe, and whose hearts find satisfaction in the remembrance of Allah: for without doubt in the remembrance of Allah do hearts find satisfaction.”
“When My servants ask thee concerning Me, I am indeed close (to them): I listen to the prayer of every suppliant when he calleth on Me: Let them also, with a will, Listen to My call, and believe in Me: That they may walk in the right way.”
Perspectives
Latter-day Saint
The truth of the Book of Mormon is known by sincere prayer. The Holy Ghost answers, often as a burning in the bosom.
God will show the truth of the book to those who ask with real intent.
Rebuttal
Sincere people of other faiths feel the same about books that contradict it. The Bible says to test the spirits. By Whitmer's account, Joseph Smith said some revelations are of man or the devil.
If the answer is right, the bosom burns; if wrong, there is a stupor of thought.
Rebuttal
The Bible calls the heart deceitful above all things. A test that gives each sincere person a different answer cannot decide between books.
The Islamic perspective
Faith rests on knowledge, not on a feeling alone. Test the book first; the heart then finds rest in the remembrance of God (﷾).
Do not follow what you have no knowledge of; the heart too will be questioned.
In the remembrance of God (﷾) hearts find satisfaction.