The point of the longer needle strikes the bobbin case and damages both the bobbin case and the needle. Surprisingly to me, you cannot hear the needle strike the bobbin case, so you don't even realize what is happening until you change the bobbin.
The other problem with using the wrong needle is that, because the point is longer, the needle could be still in the fabric when the feed dog begins to move the fabric. This can mean skipped stitches, puckering, and possibly a bent or broken needle. I can see no problem with using a 206x13 needle in a machine designed for a 15x1 needle, except that the 206x13s are considerably more expensive.
For those who like the 306 Class machines, I will try to keep a stock of 206x13 needles in my Etsy Shop, just click on the link below.