The Torah imposed the strictest of restrictions regarding Hames on Pesah; it bears the penalty of Karet. The Hachamim decreed that Hames on Pesah is not Batel (nullified) even in a mixture of a thousand times more. This requires us to act with special, extra caution and only buy Pesah products with the most reliable certification and supervision.
The Hida (Rav Haim Yosef David Azulai, 1724-1807) writes that when it comes to Pesah, all leniencies end. Rav Haim Vital is quoted as saying in the name of his master the Arizal (Rav Yishak Luria of Safed, 1534-1572) that one who is cautious to avoid even the most minute amount of Hames will not sin that year. However, I found a more precise citation in the Mishnat Hasidim by Rabbi Immanuel Hai Rikki. He quotes the Arizal as saying, "One who is scrupulous to follow all the stringencies of the Poskim and avoids eating even the most minute amount of Hames is guaranteed that the opportunity to sin will not come his way." This is more precise since it does not imply that a person's free choice can be suspended.
I believe that the source for the Arizal's teaching can be found in the following incident from the Zohar. Rabbi Elazar asked his father Rabbi Shimon Bar Yohai why the Torah was so strict regarding Hames. Rabbi Shimon proceeded to explain to him according to the secrets of the Torah. After grasping the noxious root of Hames, he asked his father, "If it's so bad, why do we eat Hames the rest of the year?" Rabbi Shimon answered that by refraining for the week of Pesah we make a Tikun that enables us to eat bread the rest of the year. We see from here that following the Halacha during Pesah grants special Heavenly spiritual assistance for the rest of the year, as the Arizal taught.
Therefore, one should avoid eating even something which has a doubt whether it is Kosher for Pesah. There was once a great Talmid Hacham who said, "I only eat that which is not kosher for Pesah!" Everyone was shocked and they asked him what he meant. He answered, "I eat potatoes-which don't need a special Kosher for Pesah stamp. I eat fruit, which doesnt need a special Hechsher. I don't eat things which may have problems."
This principle is implied in the Mishna of "Ezehu M'koman" (listen to audio for full citation) in which the Mishna states the time, place and people who eat the Korban Pesah (Pesah Offering) in a negative language, as if to say the default on Pesah is not to eat anything unless it is certainly Kosher for Pesah.
In general, it is best to minimize buying products that may have any doubt whatsoever.
