I only saw this now, but as it's an interesting question I'll give a "better-late-than-never" answer - hope it's helpful nonetheless. One major difficulty here is distinguishing between intent and outcome: The main reason given for the suppression of pre-colonial native sources and colonial writings on native customs was their "idolatrous" nature and the Christian identification of indigenous deities with the devil. Thus the official intent connects such suppressions to the christianisation campaigns esp. of the religious orders in early Spanish America. On the other hand, the outcome of burning and prohibition of sources in most cases meant the loss of invaluable information on native history and civilisation.

To get a bit more concrete regarding this suppression of information I'll first give a few examples from early colonial Mexico (16th c.). There starting with Cortés' campaigns the great majority of pre-colonial codices were destroyed, including the famous "archive" of Tezcoco. This work was continued by the religious orders, despite their studies of native cultures, so that today very few sources remain from the Valley of Mexico and Yucatán. Another factor during early colonisation were the churches' death sentences for native nobles collecting codices, serving as a deterrent to others - such penalties given out by Mexico's first arch-bishop Zumarrága were well-known at the time. Such sentences stopped gradually with the inquisition's establishment in New Spain in 1571, which held no jurisdiction over indigenous people.

This however did not mean that writings on native customs and esp. faith were let off easily. I'll quote from an answer from an earlier post of mine here:

Legislation against writings on indigenous cultures: The late 16th c. also saw the development of a stronger official animus against indigenous rights and culture – e.g. the New Laws influenced by [Bartolomé de] Las Casas were effectively undermined by the Council of Trent. A good example of this is a royal decree (Ordenanza Royal) to the Viceroy of New Spain Enríquez in 1577, which was written in response to Bernardino de Sahagún's great research into native culture and history: Enríquez was told to “in no way allow any person to write things related to superstitions and way of life that these indios had, in any language” [my translation]. Such laws have to seen in connection with the newly appointed Chronicler of the Indies' monopole of knowledge as well. Confiscations through Chronicler or Council of Indies at first often targeted works deemed too critical of the Crown's policies, especially those towards natives, and too strongly in favor of the conquistadors (or just plain unlucky). In comparison this Ordenanza Royal meant that important New Spanish works of “proto-Ethnography” by Sahagún, Dúran and other chroniclers from the orders were confiscated and often left unpublished in the Council of Indies until the 19th century. Similarly, no Spanish American author of indigenous descent (from the 16th/17th c.) apart from El Inca Garcilaso de la Vega from Peru had his works printed before the 19th century.

These were official policies initiated by the crown, in contrast to the churches' policies mentioned for the mid-16th c. I don't want to finish on such a pessimistic note though. The above-mentioned Garcilaso de la Vega (1539–1616) is an interesting though exceptional counter-example to the Spanish suppression of information. Like most indigenous chroniclers he descended from native royalty, in his case the Inca. In his well-known "Comentarios Reales de los Incas" he used stories heard from native elders as sources to describe the Incan empire's great development - even proposing their organisation as one basis of a new political model for colonial Peru (including Spanish elements as well, of course)! This chronicle circulated widely, and was seen as one inspiration for the much later Tupac Amaru II revolt in the mid-18th c. Coming back to colonial Mexico, various indigenous and native authors in the 16th & 17th cs. also highlighted the deeds and social organisation of their ancestors, including various subgroups of the Nahua, although (as I said) none were published in their life-time. On the one hand such writings can be seen as following Spanish literary models as to receive privileges and higher positions in colonial society. On the other hand authors like Fernando de Alva Ixtlilxochitl in Mexico and Garcilaso de la Vega in Peru showed that the Nahua (aka Aztecs) and Incas had societies and histories to match and partly even to exceed their European counterparts.

Let me know in case you're interested in sources other than those mentioned in the older thread I linked to.