Since the value of various objects is imputed by the mind to begin with, then there is a sense of follow-through with their disposal. For example, Some people receive stuff in the mail from Monasteries or Dharma centers and they have images of Buddhas and teachers printed on them. So, we regard these as meaningful because that is the mind.
Of course, on very close examination there are no images printed on them, merely tiny ink dots. But the mind gathers this together, and what see see is those images which in our mind are significant.
So, it does not seem appropriate to just throw it into the trash even though we know that it is just a scrap of paper. For this reason, with full understanding that this is all just play of the mind, people will sometimes burn "sacred trash" meaning old copies of teachings, maybe things that were run off on a copier but are now in published form or whatever, and, just as with incense or prayer flags, use this as an opportunity to wish that one's aspirations to benefit others are carried in all directions by the wind. So, the point is to provide an opportunity for the one disposing of the stuff.
Just as valid, of course, is the example of the raft which one discards after using it to cross a stream. Being overly attached to objects is not particularly beneficial.
It should be noted, however, that various traditions which have come to the west from the east often lose their importance due to simple cultural circumstances. As you may know, the Chinese way of writing and its development is entirely different from that of any alphabetical language. Characters are actually pictorial symbols (or derived from pictures) of things they represent. For this reason, the written word itself, as a thing in itself, has a particular value in Chinese culture that simply does not exist in the west, and not merely in terms of any calligraphic aesthetic. In traditional Chinese culture, a written character was often regarded as retaining somewhat the essence of the thing it represented. Likewise, Seed syllables such as "OM" written in Sanskrit held a similar power, or characteristic, in Indian culture.
Alphabetical languages, by contrast rely on strings of letters which represent vocal noises rather than the objects being spoken, or written about. So, for example, a chair is not a combination of cheese and hair. This may seem silly, but it is an essential consideration. But in Chinese, written characters are often combinations of things. In a sense, each character is like a charm which produces a communicated idea, and so the written word holds a different status in many Asian countries, simply due to the fact of its development.
This doesn't mean you need to do one thing or another with whatever you need to get rid of, but i just wanted to suggest that what often seems like some stupid concern sometimes has a reason which is not really so stupid after all. The reasons why buddhist garbage is often disposed of "mindfully" has its roots in history, even though the original reasons may not apply today.
When you delete files from your computer, where do they go?
Profile Picture: "The Foaming Monk"
The Chinese characters are Fo (buddha) and Ming (bright). The image is of a student of Buddhism, who, imagining himself to be a monk, and not understanding the true meaning of the words takes the sound of the words literally. Likewise, People on web forums sometime seem to be foaming at the mouth. Original painting by P.Volker /used by permission.