Turning off the game shouldn't affect your chances of getting a shiny with Masuda method. It may affect the Random Number Generator but the chances of getting a shiny won't change because of that.
The full quote from Bulbapedia is "It will not work if both Pokémon are from the same foreign language, even if it is a different language from the game; or if the player uses one of his or her own Pokémon and the Meister's Magikarp or Lt. Surge's Pikachu, which are obtained through in-game trades." That means Masuda Method won't work if both parents are Japanese or both parents are Italian. You need to have two different languages like Italian and Japanese or English and Japanese. You can't use foreign Pokemon from in-game trades because they were generated in that game and would be counted as the language of that game.
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