How dangerous would it be to start a jet engine with coins inside?
As you say "it depends". Modern jet engine front turbines are designed to reject foreign object debris outside the core, which is where the damage would really matter. Jets suck up hard foreign object debris like pebbles often enough and it will ding things a but but hardly cause catastrophic damage. I'm sure coin throwing has gone unnoticed, and it made no difference in the end. 

Really, it's not a big risk for the most part, modern high-bypass turbofans have a large bypass duct, when coins are thrown into an engine which is not running they will drop to the bottom of the bypass duct and get blown harmlessly out the back when it's started. However, there's just enough risk that a coin could find its way into the compressors and cause enough damage to need expensive repairs, and that is why they will take the airplane out of service. 

Of course it depends on the coin, if you threw this bad boy in it would be a different story: