Is the German word "blass" related to the French word "blasé"?

The contemporary meaning of a word often has very little to with its etymology. ( German "Zaun" means "fence", and its English cognate "town" means something else altogether nowadays)

I found a French site with the etymology in it. It says "of doubtful origin", but it denies the connection of "blasé" with the English "blaze" as in fire.

Now I think there might still be a (more tenuous) connection, the German "blass" is from Old High German "plass/plaess", an adjectiv which also could describe the pale ash colour of a burnt patch, and it's connected to modern German "Blesse" a noun for a white patch on an animal's (fore)head, cognate to the English "blaze".

So "burnt out" might well have been one of the connotations of "blasé".

Contrary to common perception, French has a good deal of words in its lexis which are of Germanic origin, due to the occupation of the country by the Franks in the 6th century.

I copied the French text for you:

"Blaser

Nature : v. a.

Prononciation : blâ-zé

Etymologie : Origine douteuse. On trouve, dans Du Cange, blas, sot, dépourvu de sagesse ; mais le sens ne se prête pas à la dérivation ; il n'en est pas de même de l'anglo-saxon blase ou blaese, brandon, anglais, to blaze, brûler, bas-latin blaserius, incendiaire. Le sens propre de blaser paraît être brûler ; c'est celui que lui donne St-Simon dans l'exemple ci-dessus rapporté ; et dans plusieurs provinces blaser est un terme pour signifier brûler, dessécher, lorsque cet effet est produit par l'usage excessif des liqueurs fortes."

German blass : pale

French blasé : unexcited

As a rule they wouldn't have a common root in an older language, as the languages evolved separately. French evolved from Latin, and German evolved from the germanic languages of the Hun. However, they have similar meanings and as languages evolved different groups did borrow from each other, so it's entirely possible one culture or the other borrowed and changed it.

They seem completely unrelated, the German word means pale, and the French one "satiated."

Not much in common there. There is a German word, "blasiert" with the same meaning as the French one and which probably comes from it.

Malheureusement