14ft Ceiling beam sagging 5/8" - what to do?
Without destructive removal of finishes, it is impossible to know if it is a problem because the deflection of 0.625" (5/8") on a span of 14'-0" is less than the maximum allowable deflection (length/240) that has been typical in most US building codes for many years.

The reality is that describing construction as 'meeting the code' sets the bar at the worst building the law will allow. As a practical matter, this means that floors in residences often 'bounce' and that deflection becomes visible as spans increase. However, so long as the construction meets code minimums, it does not mean that the building is unsafe.

I am:


not saying the building is safe. 
not saying the building is unsafe.


The proper way to determine the safety is to have a Professional Engineer investigate actual conditions on the site. This is really beyond the skill of a home inspector - their job is to say "the beam is sagging".