Current consensus on the Messinian Salinity Crisis
According to palaeoclimatic records modelled from sediments studied at Monte dei Corvi, in the paper Reconstructing the temperature and salinity of the Mediterranean Sea through the Late Miocene (13 Ma - 6 Ma) prior to the Messinian Salinity Crisis (Alexandrina et al. 2014), that using alkenone paleothermometry, there was a cooling trend starting at 8.1 million years ago, developing a cold period around 7 million years ago and continuing to the Messinian Salinity Crisis (MSC) as evidenced by a disruption in marine sedimentation (and changes to the surrounded flora and desertification in the Sahara).

Research presented in the paper The Onset of the Messinian Salinity Crisis along the offshore Betic Range (Ochoa et al. 2014) suggest that tectonic processes along the offshore Betic Range in southern Spain may be a trigger for the closure (see tectonic map below for the relevant area).



Image source: Wikipedia

The MSCis described in the paper Modelling global-scale climate impacts of the late Miocene Messinian Salinity Crisis (Ivanovic et al. 2014) as being the result of the influences of climatic change as observed in sedimentary strata described by Alexandrina et al. 2014 and of tectonic activity of the offshore Betic range, as detailed by Ochoa et al. 2014.  They also suggest that during the MSC, there were intermittent outflows of saline water into the Atlantic - causing disruptions to the thermohaline circulation.

Dr. Iuliana Vasiliev of the University of Utrecht, suggests that one possible influence during the MSC was interactions with the closing remnants of the Paratheys Ocean, as shown in the map below: