Ali ibn Ibrahim has narrated from his father from Ibn Abi ‘Umayr from Hammad from al-Halabiy who has said the following: “I once asked abu ‘ Abd Allah (a.s), about a man who chews chewing gum when fasting. He (the Imam) said, ‘No.’” (Al Kafi)
Muhammad ibn Yahya has narrated from Muhammad ibn al-Husayn, from Ali ibn al-Hakam from al-‘Ala’ ibn Razin from Muhammad ibn Muslim who has said the following: “Abu Ja‘far (a.s), once said to me, ‘O Muhammad, never chew chewing gum because I today did so when I was fasting and I found something from it in myself (i.e. particles in my mouth).’” (Al Kafi)
These hadiths explain eachother. The second hadith explains that the reason for the prohibition of chewing gum is because there may be particles left in the mouth after chewing. There is no problem in chewing, however, if particles would not be left in the mouth, or if one does not have an expectation that particles would be left in the mouth.
As such, Sayyed Sistani's website has the following question and answer:
Question: What is the ruling about chewing gum in the month of Ramadhan?
Answer: There is no problem in chewing a gum though he might feel the flavor insofar as the gum does not have particles to be swallowed. But if it has particles which spread in the saliva, it is necessary to avoid chewing it or swallowing the saliva which contains the particles. In case there are no particles or substance to be swallowed, there is no problem in chewing it. (Sistani.org)
Objection 1: The first hadith is categorical, so it cannot be contradicted.
Response: The Quran speaks in a similar fashion, in 2:123, the Quran categorically says there will be no intercession: "And fear a Day when no soul will suffice for another soul at all, and no compensation will be accepted from it, nor will any intercession benefit it, nor will they be aided." (2:123). However, in 20:109, the Quran shows an exception: "On that Day intercession shall not avail save of him whom the Most Compassionate Lord permits, and whose word of intercession is pleasing to Him" (20:109). So there is no problem if the Imam gives a categorical and general answer in one hadith, and specifies conditions in another hadith.
Objection 2: If the Imam chewed gum, this violates the Imam's knowledge of Ilm al Ghayb, since he should have known the gum he chewed would have released particles to be swallowed.
Response: Our belief of the Ilm al Ghayb of the Imams is the same as the Ilm al Ghayb of the Prophets.
Ibn al-Mughairah who said: Yahya ibn Abdullah ibn al-Hasan and I were with Abul Hasan, peace be upon him, and Yahya asked him: "May I be your ransom, they think that you have the knowledge of the unseen (Ilmul Ghaib)." He said: "Glory be to Allah! Place your hand over my head." When I did that, every hair in my head and on my body stood on its end. Then he said: "No, By Allah, it is nothing but what we have inherited from the Prophet (peace be upon him and his progeny)." (Amali al Mufid)
Hence we can draw a parallel from the Quran and say that the example of the Imam chewing gum is the same as the example of Prophet Moosa (a) punching a man and killing him by mistake:
And he entered the city at a time of carelessness of its folk, and he found therein two men fighting, one of his own caste, and the other of his enemies; and he who was of his caste asked him for help against him who was of his enemies. So Moses struck him with his fist and killed him. He said: This is of the devil's doing. Lo! he is an enemy, a mere misleader. (28:15)
Obviously, Moosa (a) did not intend to kill the man with his strike, and similarly, it can be the case that the Imam (a) did not intend to have chewed the type of gum which would have left particles in his mouth. As such, there is no problem and no sin committed if the Imam does something which is presumed lawful, but it turns out to be a mistake in the same way Moosa (a) did something lawful, but it turned out to cause an error.
Similarly, the Imam had married a nasibi woman, but when he later found out she was a nasibi, he divorced her:
“Abu Ja‘far (a.s), has said, ‘Once a man came to Ali ibn al-Husayn (a.s), and said, “Your wife from al-Shayban people is of Kharijiy sect who vilify Ali (a.s). If you like I can manage for you to hear what she says.” He (the Imam) said, ‘Yes, I agree.’ He (man) said, ‘Tomorrow as you come out of your home as you normally do, wait on the side of the house.’ Next day he (the Imam) waited on the side of the house and the man came. He spoke to her and it (vilification) became clear from her. He (the Imam) divorced her although he liked her quite well.’” (Al Kafi)
So the Imam chewing gum while fasting does not invalidate his infallibility in any way.