I installed now the official backport of calendar-exchange-provider and tested again.
Result: With icedove 31.7.0-1~deb8u1 everything works fine, i.e. there is no password dialogue at startup nor later. With Icedove 38.5.0-1~deb8u1, I get asked for the exchange password twice(!?) at every start of Icedove. Afterwards it works until the next start of Icedove. On my mixed testing/unstable system, everything works fine! Because this is now only an annoyance, I suggest to downgrade the bug to "normal" (or even to "minor"). Btw. many thanks for the official backport, Mechtilde! Very much appreciated!
