As some of the comments allude to, a lie is generally considered a communication with an intent to deceive. WP uses this definition:
A lie is an assertion that is believed to be false, typically used with the purpose of deceiving someone.
As such, if person B is relaying with good faith a communication from person A, they are not lying because they most likely have themselves been deceived, although it is arguable that person B has ethical obligations to verify the communication from person A. For instance, if person B is a newspaper reporter and mindlessly passes on a claim, then they may very well be in violation of journalistic ethics. This notion of the difference between those who intentionally and unintentionally pass false information is covered by a distinction between misinformation and disinformation. Note that if person B is doing a favor for person A, then B's noble intentions may not absolve A from ethical responsibilities. A classic proverb is "The road to hell is paved with good intentions." Thus, a ethical consequentialist will ultimately judge person B and their ethics by the outcome of spreading misinformation.