It is commonly believed that Tolkien disliked, or had strong opinions of, William Shakespeare. However, there seemed to be a lot of nuance.
We would see that there were some plays he enjoyed and some he gave criticisms of.
One which he enjoyed was a live play of Hamlet calling it a “very good performance” (Letter 76)
However, it would be shown in other letters he would despise certain aspects of Shakespeare’s work in Macbeth and others.
However, the latter could also be seen as a form of literary criticism, critiquing aspects of his works while not projecting an overall view of Shakespeare as a whole.
Inspirations
Meanwhile, Tolkien wrote of his “bitter disappointment and disgust from schooldays of the shabby use made in Shakespeare of the coming of the ‘Great Birnam wood to high Dunsinane hill’” in Macbeth, attributing to his creating of Ents such as Treebeard – wanting a setting where trees might rise to war, and the part where Éowyn defeats the Witch-king is known to be inspired by Shakespeare’s Macbeth and King Lear.
Some might say it was a ‘schoolboy’s dislike,’ though he did use it to one day create a setting of Ents. It is unknown the exact link to those and his works, though many have speculated that it was Tolkien’s desire to create something Shakespeare did not. However, that could just be mere inspiration, or literary criticism at most, and not an indicator to his full opinion of Shakespeare’s works.
Hamlet
“Plain news is on the airgraph; but the only event worth of talk was the performance of Hamlet which I had been to just before I wrote last…it emphasized more strongly than anything I have ever seen the folly of reading Shakespeare (and annotating him in the study), except as a concomitant of seeing his plays acted.”
Thus he would volunteer the information that he had viewed a live play, showing some strong emotion. It also seemed like while he had an issue with one particular part, he enjoyed the play at least for its own sake. Tolkien also seemed to be able to recall and enjoy a part he did not expect to enjoy.
“It was a very good performance, with a young rather fierce Hamlet; it was played fast without cuts; and came out as a very exciting play. Could one only have seen it without ever having read it or knowing the plot, it would have been terrific. It was well produced except for a bit of bungling over the killing of Polonius. But to my surprise the part that came out as the most moving, almost intolerably so, was the one in reading I always found a bore: the scene of mad Ophelia singing her snatches.” (Letter 76)

This might seem to be contradictory to his quote in On Fairy Stories, where he describes talking animals.
“Fantastic forms are not to be counterfeited. Men dressed up as talking animals may achieve buffoonery or mimicry, but they do not achieve fantasy.” (On Fairy Stories)
While this is commonly attributed to Shakespeare’s use of elves and fairies, it’s hard to tell whether Tolkien was referring to Shakespeare, or other works. After all, he had referenced to Shakespeare in the same work, but not entirely related to this quote.
Elves in literature
Beliefs that Tolkien disliked Shakespeare came from what he said about Elves. Much like Tolkien, Shakespeare also incorporated Elves in literature. However, they were shown differently. Thus it is commonly believed that Tolkien wrote his elves to counter Shakespeare’s, as evident by the quote:
“I now deeply regret having used Elves, though this is a word in ancestry and original meaning suitable enough. But the disastrous debasement of this word, in which Shakespeare played an unforgivable part, has really overloaded it with regrettable tones, which are too much to overcome.” (Letter # 185).
Elves started to be a prominent form of literature from the early modern period, which Shakespeare was in or 1500-1700. However, what set Tolkien against Shakespeare in this sense was taking away the distinctive character of the elves.
Tendencies
Much like what we would see in his views on other works, we would see again that he had two tendencies: hyperbole (as an element of acting) and understatement (which was similar to his neighbors in England). This was cited by friends such as C.S. Lewis as well – and part of his personality.
CS Lewis once remarked that “Tolkien’s lively mind sometimes leads him (with perfectly innocent intentions) to overstate things.” Christopher Tolkien noted that his father had a tendency toward the “rhetorical superlative.” John Garth has identified it as a habit of “mischievous hyperbole.”
Tolkien’s Modern Reading, Ch 12
Thus criticisms such as this one on Shakespeare, as well as Disney, George Macdonald, Hans Christian Andersen, and others – must be read with this habit in mind.