I don't know about innapropriate, but I'm not sure how many of the tunes I've considered as appropriate in my more morbid moments (Of which I don't have many, surprisingly) would be allowed to be played in a place with any mainstream religious pretensions. I refer to songs like Cryptic Wintermoon - When Daylight Dies, Xasthur - The Funeral of Being, Coil - Tainted Love, Sol Invictus - Come Join the Dance, Inkubus Sukkubus - Leveller, Burzum - Tomhet, Ewigkeit - Journey to Ixtlan, etc. etc.
Some psychobilly would be awesome at a funeral, but I'd probably go for some Coffinshakers ballads.