Answer
No, causality does not have to be discarded.
Work Step by Step
If event A causes event B, then it must be possible to connect the 2 events by a beam of light.
In that case, for frames moving relative to each other at speeds less than c, the time interval between the 2 events is measured to be different, but the order in which the events occur is agreed upon by all observers.
This is a large topic that this question barely touches upon. The 2 events are known as timelike events.
