What is the ruling on a man completely ignoring a non-mahram woman and vice-versa, a woman completely ignoring a non-mahram man, out of fear of temptation, such that if he or she greets or asks a question, they are ignored and shunned, even if they are a relative?

When there is a fear of fitna (strife/temptation), speaking between a man and a strange young woman is prohibited, even if it is merely exchanging greetings, because it is a place where fitna is suspected. However, if there is a need to speak or greet with the assurance that there is no fitna, then it is permissible, such as the presence of a kinship that prevents abandonment and estrangement.