This is one of the biggest lies in the history of mankind and literally all evidences from Islamic sources speak against it starting from the Sirah to mass narrated narrations (if you doubt all of them then you can go ahead and doubt the Qur'an since it was narrated by the very same NON-INFALLIBLE people).
Historical accounts testify that the Prophet (SAWS) sent non-infallibles to lands as far away as Yemen, to do what? To sell Shia praying-pucks (aka Turbah)?! No, rather to teach them the pillar of Islam itself (Tawhid) as we can see in the Hadith of Mu'adh Ibn Jabal (a non-infallible) who was sent in the name of the Prophet (infallible) to teach the people the Deen (the core of the Deen):
The people of Yemen took the Deen from (at least) two fallibles who obviously narrated what they knew from the last infallible of this Ummah (the Prophet) i.e. exactly what the Ahl Al-Sunnah do when they narrate hadithNarrated Ibn Abbas (Hashimi from the Ahl Al-Bayt himself):
When the Prophet
sent Muadh to Yemen, he said to him, "You are going to a nation from the people of the Scripture, so let the first thing to which you will invite them,
be the Tauhid of Allah. If they learn that, tell them that Allah has enjoined on them, five prayers to be offered in one day and one night. And if they pray, tell them that Allah has enjoined on them Zakat of their properties and it is to be taken from the rich among them and given to the poor. And if they agree to that, then take from them Zakat but avoid the best property of the people."
(Sahih of Imam Al-Bukhari, Volume 9, Book 93, Number 469)
^Note how the Prophet (as in ANY other hadith) did not mention anything about following 12 particular individuals from Ali's ofspring, yet he trusts a non-infallible (who is amongst the Sahaba who gave bay3a to Abu Bakr) to teach the people about Tawhid and Islam.
Another non-infallible (heck not even from the Bani Hashim!) who was sent to his homeland to teach the people about Islam is Abu Musa Al-Ash'ari (whom the Rafidah hate with passion, because of the fitna he was involved against Ali):
Narrated Abu Musa Al-Ashari:
Allah's Apostle sent me (as a governor) to the land of my people ... [long hadith]
(Sahih of Imam Al-Bukhari, Volume 5, Book 59, Number 633)
And there are other such instances where the Prophet (SAWS) sent non-infallibles (although Ali is alive and has super powers acc. to Shias and can be at any place and any time acc. to Shias) to teach the people Islam and the Qur'an.
Now here is the question: How can the Prophet (SAWS) do such a thing if Twelver Shiism claims that only an infallible understands the Qur'an? How could those non-infallibles possibily explain Islam, Tawheed and the Qur'an to the people if they themselves were not just non-infallibles but in fact future apostates such as:
Abu Musa Al-Ash'ari who is a Nasibi Kafir apostate acc. to Twelver Shiism. Their stance is well known on him
Mu'adh Ibn Jabal who he has been promised hell according to this world famous Shia "Center of Belief researches" websites, based on their narrations that say that Mu'adh was promised Hell for chosing Abu Bakr and Omar (i.e. giving them Bay3ah) over Ali:
http://www.aqaed.com/faq/6294/ The answer of course is that they were neither apostates or Nasibis nor does Islam require that only an infallible explains the Qur'an and this as per the noble Qur'an itself:
وانزلنا اليك الذكر لتبين للناس ما نزل اليهم
"We have revealed upon you the Zikr , so that you make clear
to the people what we have revealed for them..." (Surah An-nahl ,Chapt 16 verse 44)
Also:
لقد من الله على المؤمنين اذ بعث فيهم رسولا من انفسهم يتلوا عليهم ايته ويزكيهم ويعلمهم الكتب والحكمة.
"Verily , Allah favoured the believers when he raised a messenger amongst them. He recites his verses upon them
and purifies them and teaches them the book and wisdom..."
Surah Aale-Imran , Chapt 3 verse 164
THEM! His companions. If we were to be as ignorant as the Shias then we would claim that the companions are infallible since Allah is PURIFYING them. In any case, they are being TAUGHT and obviously they will teach future generations (and not just "12 Imams" who altogether live less than 250 years, hence the Shia Ummah itself is without an infallible guide for over 1200 years!).