Dryden writes that Shadwell is "thoughtless as monarch oaks, that shade the plain, / And, spread in solemn state, supinely reign." Firstly, he means by this simile that Shadwell is unthinking and lacking in dynamism, just like a tree that is rooted to the same spot. Dryden also describes how the tree is "solemn" and seems to reign "supinely." The word "supinely" connotes passivity, or inactivity, which, in combination with the word "solemn," seems to suggest that the tree (and thus Shadwell) appears somewhat pretentious and self-important. The oak tree likes to "reign," but does so lazily. In the same way, Shadwell likes to assert his own importance (he wanted to be known as the heir to the great English poet, Ben Jonson), but does so perhaps lazily, or complacently.
When Dryden compares Shadwell to a "monarch oak," it is also possible that he is being sarcastic. Shadwell considered himself the next branch of a great English poetic tradition. Indeed, as mentioned above, he considered himself the successor to Ben Jonson. Possibly from his own perspective, Shadwell might have thought that he had much in common with the "monarch oak." The oak is strong and impressive and a symbol of England. Its roots go deep into the earth, just like Shadwell thought his roots, as the successor to Jonson, also went deep into English tradition. Dryden is thus possibly mocking Shadwell's over-inflated sense of self-importance in this comparison.
When Dryden subsequently accuses Shadwell of being the "last great prophet of tautology," he is attacking Sheldon's craft as a poet. A tautology is an uneconomical, ineloquent use of language, and so quite a criticism of someone who claimed to be a wordsmith. The sarcastic tone of "last great prophet" also supports the above interpretation of the oak tree simile as a sarcastic riposte to Shadwell's own self-importance.