In this sentence, is summer a noun even though it is a direct object?
Go during the summer.
Yes. Summer is a noun.
Direct objects are always nouns or pronouns.
I think you're confused about the 8 parts of speech (noun, pronoun, adjective, adverb, verb, preposition, conjunction, interjection) and parts of a sentence (subject, verb, direct object, etc.)
http://www.uottawa.ca/academic/arts/writcent/hypergrammar/grammar.html
You can get explanations about the parts of speech (noun, pronoun, etc.) and the parts of sentences (subject, verb, etc.) by using the list at the left.
http://www.chompchomp.com/menu.htm
And here ... in the Terms section.
http://members.cox.net/lenco1/grammarpractice/
And here ...
Yes, in the sentence "Go during the summer," the word "summer" is a noun. Even though it is functioning as a direct object, it still remains a noun.
To identify that "summer" is a direct object, you need to understand the sentence's structure. In this case, the verb is "go," and "summer" is the receiver of the action. The question "Go when?" is answered by the noun "summer." That's why "summer" is considered a direct object in this sentence.
To verify this, you can ask the question: "What are you going during?" The answer is "summer," which confirms that "summer" is indeed the direct object.