I am an inhabitant of Muslim country though Muslims are not much conscious here. In our country traditionally "Bismillahi al-Rrahmani al-Rrahim" is used to be written at the top of poster, banner, festoon, pamphlet, etc. It does not matter but the problem arises when these are thrown in street where people loiter without much caring for. If I find such poster or pamphlet should I tear the portion where "Bismillahi al-Rrahmani al-Rrahim" is written? It can be done with ease if only 1 or 2 is thrown away but when numerous of these are scattered in street what to do?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

One can get rid of papers that include the Name of Allah or Quranic verses by burning them or wetting them and then burying them, or by special modern machine (shredder) that destroys papers.

‘Uthman Ibn 'Affaan ordered the burning or cutting into pieces of the old sheets on which the Quran was written after collecting them and compiling their contents into a complete copy.

In the same vein, the above way of getting rid of papers that include Allah's Name/Quranic verses is also reported from 'Urwah Ibn az-Zubayr and Taawoos .

But, if there is a large quantity of such papers, then one can do as mentioned above and according to his ability.
Allah Says (what means): {So fear Allah as much as you are able.} [Quran 64:16]

After that, you have to do your best to make Muslims aware of the issue that you raised and possible solutions: Burning/burying papers that include Allah’s Name or Quranic verses.

For more benefit, please refer to Fataawa 352552, 238213, 87024, 263416, and 338455.

Allah knows best.