Why don't the Amish like their picture taken?

Exodus 20:4 says "You shall not make for yourself an image in the form of anything in heaven above or on the earth beneath or in the waters below" (NIV).

In the latter part of the 19th century, as photography was becoming popular, the Amish applied the biblical injunction against "likenesses" to photographs.  (From The Riddle Of Amish Culture by Donald Kraybill).

They also believe that photos of themselves are prideful; these are a people that believe in community and look down upon individualism.  Pridefullness leads to arrogance, which in turn leads to self importance, quite the opposite of community.