Does Scrabble allow playing words which contain diacritics in their original spelling?
Scrabble does indeed allow words of a foreign origin IF they are used consistently enough in the English language to appear in an English dictionary.

Generally speaking this would result in the spelling being anglicised to remove diacritics since they have no meaning in English. As such you are highly likely to find the words naive, nee, cafe in an English dictionary without diacritics, thus they are usable.

From the Scrabble rules (at the end under Accepted Scrabble Words):


  When playing an English version of the game, foreign words are not allowed to be placed on the board. However, if the foreign word does appear in a standard English dictionary, it is allowed. The reason for this is due to the fact that the word is spoken enough and is considered part of the English language.


It is also worth noting that even in the French edition of Scrabble diacritical marks are not present in the letters and are merely inferred by placing the appropriate unmodified letter.

The Wikipedia entry for Scrabble letter distribution also specifically states that diacritical marks such as é are ignored in words borrowed from French.