Lots of people in our parish hold hands during the Our Father at mass. I do not. I typically fold my hands in front of me, even when our priest, from the altar, invites us to “Join hearts and hands.” He has no right to do that.
If people want to spontaneously join hands, that’s fine. It’s not part of the rite, though, and Father has no authority to add it. Lest those around me think that I’m a grumpy curmudgeon, I find it necessary to go out of my way to offer a smile and a firm handshake during the sign of peace immediately following the Our Father.
It’s nice to know that Fr. Z of What Does the Prayer Really Say pretty much shares my view.
There is no specific prohibition against holding hands during the Our Father, or any other time at Mass for that matter. However, there is no provision to ask or invite people to do so, and were a priest or deacon to do so during Mass he would be committing a grave liturgical abuse. Priests can’t just make stuff up and impose things because they think it is meaningful.