The standard rule is that when you have multiple quoted paragraphs, each new paragraph starts with an opening quotation mark, but only the final quoted paragraph has a closing quotation mark at its end.
Example:
“That seems like a strange way to use punctuation,” Bob asked. “What harm would there be in using quotation marks at the end of every paragraph?”
“Oh, not so,” Diane replied. “If you closed quotes at the end of every paragraph, then you would need to reidentify the speaker with every subsequent paragraph.
“Say a narrative was describing two or three people engaged in a lengthy conversation. If you closed the quotation marks in the previous paragraph, then a reader wouldn’t be able to easily tell if the previous speaker was extending his point, or if someone else in the room had picked up the conversation. By leaving the previous paragraph’s quote unclosed, the reader knows that the previous speaker is still the one talking.”
“Oh, that makes sense. Thanks!” said Bob.