Is it permissible for me to communicate with my irreligious brother without going to his house or clearing our hearts, in order to avoid falling into sins and problems resulting from the dealings of his irreligious wife?

Your brother is a blood relative, and maintaining ties with him is obligatory and carries immense reward. If it is possible to visit him without any negative consequences, then do so. Otherwise, maintain ties with him through whatever means are available, for the ways of maintaining ties vary. As for your statement, "only purifying hearts," if you mean by it merely the soundness of your own inner self, then that is not sufficient for maintaining ties, especially if your brother believes that his wife's satisfaction is a condition for your maintaining ties with him, which is something neither Islamic law nor reason supports.