“The boy feels warm” is correct.  “The boy feels warmly the rebuke of his teacher” is equally correct.

While license is granted to the poets to use the adjective for the adverb, as in the line

“They fall successive and successive rise,”

in prose the one must never be substituted for the other.
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“Agreeably to my promise, I now write,” not “Agreeable to my promise.”

“An awful solemn funeral,” should be “An awfully solemn funeral.”

“He acts bolder than was expected,” should be “He acts more boldly.”

“Helen has been awful sick, but she is now considerable better.”  “Helen has been very ill, but she is now considerably better.”

Do not use coarser for more coarsely, finer for more finely, harsher for more harshly, conformable for conformably, decided for decidedly, distinct for distinctly, fearful for fearfully, fluent for fluently.

Do not say “This melon is uncommon good,” but “This melon is uncommonly good.”

The word ill is both an adjective and an adverb.  Do not say “He can illy afford to live in such a house,” but “He can ill afford.”

“That was a dreadful solemn sermon.”  To say “That was a dreadfully solemn sermon” would more grammatically express what the speaker intended, but very or exceedingly would better express the meaning.

Such, So

Such is often improperly used for the adverb so.

“In such a mild and healthful climate.”  This should be, “In so mild
and healthful a climate.”
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“With all due deference to such a high authority on such a very important matter.”  Change to, “With all due deference to so high an authority on so very important a matter.”

Good, Well

Many intelligent persons carelessly use the adjective good in the sense of the adverb well; as, “I feel good to-day.”  “Did you sleep good last night?” “Does this coat look good enough to wear on the street?” “I can do it as good as he can.”  The frequent indulgence in such errors dulls the sense of taste and weakens the power of discrimination.

Very much of

“She is very much of a lady.”  Say, “She is very ladylike.”  “He is very much of a gentleman.”  Say, “He is very gentlemanly.”

Quite

This adverb is often incorrectly used in the sense of very or rather.  It should be employed only in the sense of wholly or entirely.  These sentences are therefore incorrect: 

“He was wounded quite severely.”

“James was quite tired of doing nothing.”
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How

This word is sometimes used when another would be more appropriate.

“He said how he would quit farming.”  Use that.