Which is the correct usage - " the line is breaking off " or " the line is breaking up"?

When the other person's voice is affected by poor signal, we use 'the line is breaking up'.

If the the call is unintentionally ended because of poor signal, we use 'the line has cut off' (you could say 'the line has broken off', but idiomatically it's better to use 'cut off').

It may be helpful to think about an actual line, drawn on a piece of paper. If the line is 'breaking up', there are still little pieces of the line left, because we are in the process of breaking. If the line has 'broken off', then the process of breaking is completed - the line has reached a premature end and there are no more pieces left.