The sentence is: No one grew tired of eating it. Is there any prepositional phrases in there?

The other sentence says: Linking verbs link the subject with a noun, a pronoun, or an adjective. Is everything after with(also including with)a prepositional phrase?

Although your first sentence looks like it has a prepositional phrase, tired of is a verb phrase.
http://dictionary.reference.com/browse/tired%20of

Yes. The prepositional phrase in the second sentence is with a noun, a pronoun, or an adjective.

In the first sentence, "Of eating it "is a prepositional phrase. "of" is the preposition and "eating it" is the gerund object of the prep.