In the sentence "David is going to see her again", the complete infinitive phrase is "to see her again" and it functions as the object.
An infinitive phrase is composed of a group of words that uses an infinitive: “to” + verb. It has the role of expressing an action in a sentence. Moreover, an infinitive phrase is part of a sentence, so it usually functions as a noun, adjective or adverb.
In the case of the sentence, as the infinitive phrase answers to the question "What is he going to do?", the infinitive phrase functions as the object.
