The sutra are the sarira of the Buddha's Dharma body. The sutra contain great wisdom, knowledge and practice methodologies. Recitation of sutra, even without understanding or pondering upon the ideas conveyed in the texts, brings great benefit to one's practice in Buddhism as well as worldly gains. Where there are sutra, there tends to be Dharma protectors guarding them. We should use and handle sutra in a manner that is respectful and courteous. My own altitude is to always view and treat sutra at the same level as the Buddha's body.
In the Chinese Buddhism that I have observed, practitioners are taught early in temple practice to handle sutra in specific ways, but most of these are confined to during temple rituals. Outside such ceremonies, it seems not all Chinese Buddhists always handle sutra with the same level of care, respect and courtesy. I do not know how sutra handling practices are taught in non-Chinese traditions of Buddhism - so please share your experiences.
What I share below are things that I have learned as the proper way to use and handle sutra, in no particular order:
1. Always wash your hands clean before handling a sutra. Even better if you can take a bath or shower.
2. Do not take a sutra into places that are or are perceived as dirty or potentially contaminating, the most common example being the toilet. I think we have to use our common sense here, for example I would not take a sutra into a slaughterhouse, a brothel, a pub, etc. After going to the toilet or one of the above-mentioned places, wash your hands or even better take a shower or bath before handling a sutra.
3. If possible and practical, wear a robe appropriate for your practitioner identify (e.g. bhikṣuṇī, upāsaka) before using a sutra.
4. Always hold a sutra with two hands.
5. Never hold or carry a sutra below your waistline. When not in use, place them in your altar or store them in a clean place that is above an adult's waistline. Some temples are very crowded, with no space for desks or benches - they may place sutra on the seats as a convenient means even though that would be below the waistline. It is very important that one does not sit on or step across a sutra.
6. When storing a sutra, you should not stack on top of it another book that is not also a Buddhist sutra. Try to keep them away from materials that are not very Buddhist in nature, e.g. porn.
7. Take good care of sutra. Any damage should be carefully repaired, e.g. with sticky tape, binding materials. Do not make markings on a sutra, e.g. with a pen, highlighter. If you have to write notes when studying a sutra, keep them on a separate piece of paper or booklet.
8. When a sutra has become so old and worn that it is no longer suitable for reading, do not destroy, throw away or burn it. The safest thing to do is to pass it to a temple for their handling or storage. There are proper methods to burn them with a special container and the ashes has to be disposed of in specific ways, e.g. sprinkled into the ocean. The burning process may involve rituals, recitation of mantra and contemplation, best done by a monk or nun.
9. With other media of storing sutra, e.g. CDs, electronic copies downloaded from the Internet, audio recordings, the same level of care should be exercised. This can be tricky as this is a modern thing where there are little guidelines from monks and nuns in past decades. One idea I have come across is to store all electronic copies of sutra into one single removable USB drive, save nothing else in that drive and plug it into your computer for access only when you read the sutra.
10. Before reciting a sutra, one should chant Śākyamuni Buddha's name (e.g. "Namo Śākyamunaye Tathāgatāya", usually three times) and preferably also a "sutra-opening verse". When reciting a sutra, one should sit upright and not be doing other things, e.g. eating, fidgeting. This may be slightly relaxed when studying (instead of reciting) a sutra, but still maintaining an attitude of respect and courtesy. After reciting a sutra, it is best to transfer and share the merit. If you cannot recite the whole sutra in one go, as is often the case with the longer sutra, you may put it down and later pick up from where you left.
This is what I can recall for now.