When jet aircraft fly into a dense cloud of volcanic ash, the engines can sustain damage, flame out, potentially causing a crash. Less dense clouds can cause windshield pitting, paint damage, melted rock deposition, and engine compressor blade erosion.
Only twice in the history of aviation have commercial jet engines failed due to ash. Once was in 1982, when a British Airlines 747 (tag G-BDXH) bound for Jakarta Indonesia lost all engines while flying through a thick plume from Mount Galunggung. The other occurrence was in 1989 when a KLM 747 (tag PH-BFC), on its way from Amsterdam to Tokyo, lost all four engines flying through an ash cloud from Mount Redoubt while descending into Anchorage Alaska.
In both occurrences, engines were restarted and the aircraft landed successfully.
KLM PH-BFC in recent years
G-BDXH in 2003 after being sold to a charter outfit
The disruption in European air travel is greater than that seen following the 11 September 2001 attacks when flights were only shut down for three days in Europe. Estimated losses by the airlines are $200M per day, plus of course the untold millions lost because business travelers are stranded and can't do business (although I saw that the President of Iceland was running his country via an iPad).
The airlines want to fly. They have stated to regulators that it is safe to fly, and they have performed test flights over the last few days to prove it. But the governments won't allow it.
It all seems rather silly. Reasonably good-sized eruptions occur in Iceland every 2-3 years, with large ones every 5-10 years. In the past, aircraft have been routed around the denser clouds. For example, in 2004, Grímsvötn erupted and caused ashfall in Europe. No closures resulted.
When Mount St. Helens exploded in 1980, I saw significant ashfall in Colorado, more than 1000 miles away. (A layer of fine dust was visible on cars and other surfaces, with some particles up to a millimeter in size.) It was a huge eruption, much larger than the current one in Iceland. Airspace closures only affected the areas right around the eruption -- and I know personally that research aircraft continued to fly directly into the plume (but not without some damage).
If this goes on much longer, airlines will start defaulting on debt and may require government bailouts or nationalization.