What is this INOP instrument and when is it safe to fly with INOP instruments?
It is a standby display, specifically a DDRMI (Digital Distance Radio Magnetic Indicator).  

The aircraft can be dispatched with inoperative equipment. The process is controlled by a Minimum Equipment List (MEL). The operator will have an MEL for each type of aircraft in their fleet. The MEL is approved by the regulatory authority (FAA, CAA, etc.) and specifies what may be inop and associated restrictions.  

I don't know what the BA MEL for the A319 is, but here's a snip from the FAA's Master MEL (MMEL), which is the guidance for each operator to use when creating their MEL.  



Following this guidance, the MEL would say you can dispatch with the DDRMI INOP if the FAR equipment minimums are met. That (at least in the US) is along the lines of "navigational equipment necessary for the planned flight".