I heard about this terrible take on empathy as sin. The person promoting that and the group Desiring God (founded by John Piper) are an ultraconservative branch of evangelicalism. More fundamentalist than evangelical. There was quite a bit of pushback on it. Scot McKnight who is an evangelical professor and blogger wrote an objection as did others:
This is not the first on this problem though. I recall this type of thinking growing up in the Church. It is not what you find in mainstream Christianity though--only ultraconservative. Rachel Held Evans a highly popular blogger wrote about this topic several years ago (unfortunately she died prematurely a couple of years ago). She comes for an evangelical background:
The empathy conversation is very gendered in conservative Christianity. I was always taught that men are naturally inclined to justice (which requires suspending compassion and implementing punishment--justice equated with judgment), while women are inclined to compassion and so women cannot be trusted to step up and do what is just at the end of the day because they rely to much on their feelings.
I haven't kept up on all the conversation, but it seems like a misunderstanding of what empathy is.