Filtering undrinkable liquids within the process of drinking:
Filtering wormy water while drinking: It is permitted to drink it through a cloth napkin [such as by placing it on ones mouth and then pouring the water through it, or by placing the napkin on the opening of the bottle and drinking directly from the opening].
The reason for this is: because the separating and filtering prohibition only applies when one is mending [the food] prior to eating or drinking it in order so it be able to be eaten or drank [afterwards] as doing so is a form of an [individual] action [of separating], however when holding back the waste upon drinking in order so it not enter into ones mouth is not similar to an action [of separating] and is [thus] permitted.
The reason that doing the above does not contain a laundering prohibition: Doing so is not forbidden due to [the law] that the soaking of a cloth is considered laundering and neither due to a decree that one may come to squeeze it, being that there is only a small amount [of cloth] which is soaking in the water [as it is only] the size of the opening of his mouth, and by a small amount there is no worry [regarding the above] as explained in chapter 302 [Halacha 21].
Other Opinions: However according to the [dissenting] opinion there which prohibits [soaking in water] even a small amount [of cloth] due to [the law] that the soaking of [even a brightly clean/white] cloth is considered laundering [as although one does not intend to launder it, nevertheless this is an inevitable occurrence], here too [according to their opinion filtering water that contains worms through a cloth napkin] would be forbidden [even though only a small amount of the napkin is getting wet, which is the area that is placed by the mouth].
The Final Ruling: It was already explained there [in chapter 302] that every meticulous person should be stringent upon himself when possible. [See Q&A]
In chapter 302/21 Admur stated this ruling as follows: One need not protest against those which are lenient [to filter through a cloth when placed by ones mouth] as according to the first opinion this is permitted in all cases [even when done with intent to launder], however every G-d fearing Jew is to be stringent upon himself not to enter into a questionable Biblical prohibition.
Summary-Filtering a liquid in the midst of drinking it:
Placing a filter on the opening of one’s mouth and then pouring in the liquid, is permitted under all circumstances, even if the liquid is undrinkable in its current state. It is allowed even if one is using an undesignated white cloth. However every meticulous and G-d fearing person should avoid, when possible, filtering white liquids when using an undesignated cloth [see Q&A], due to opinions which rule doing so contains a laundering prohibition.
Practically is only a “meticulous” person to avoid being lenient or every G-d fearing person?
In Rav Farkash’s summaries he seems to side that one is to be stringent and that only in a time of need may one be lenient.
However the Ketzos Hashulchan does not seem to accept this being that he simply rules that “it is allowed and a meticulous person is to be stringent” thus implying that one may be lenient if he so wishes.
Need one be stringent when filtering water with a cloth that is designated to be used for the purpose of filtering?
Is there any difference in the above laws if the cloth is colored or white or if made of wool or linen?
There is no difference.
May one who follows the lenient opinion filter even through a dirty cloth that is placed by ones mouth?
One should try to do so only with a very clean white cloth being that in such a case there is not much suspicion the cloth will whiten further.
May one use his shirt sleeve as a filter?
Many Poskim have written against doing so even if one places the sleeve against his mouth, as there is worry one may come to squeeze his sleeve from the absorbed liquid.
May one who is stringent nevertheless filter colored liquid through a cloth while drinking?
Yes. This is allowed according to all as by colored liquids the laundering prohibition does not apply.
May one drink a liquid that contains solids through a bottle which filters out the solids? 
Yes. This may be used according to all being that there is no laundering prohibition applicable here.
May one use a filtering straw on Shabbos to drink liquids which need filtering? 
If the filter is positioned in the area of the straw that enters one mouth then this is permitted to be used according to all. However if the filter is on the lower part of the straw then this matter is questionable. Practically in such a case one is to be stringent although one who is lenient has upon whom to rely.
 There the following is explained: There is a dispute whether placing a minute amount of water on a cloth is forbidden. The first opinion holds that it is always permitted by a clean and white cloth, even when done intentionally to launder being since it is already clean laundering is inapplicable, and thus the only suspicion in such a case is that one may come to squeeze out the water and by a small amount of water this suspicion does not apply. The second opinion holds that it is always forbidden due to that even the soaking of a white cloth is itself considered laundering and is thus forbidden in all cases even when done not for the intent of laundering being that the laundering is an inevitable occurrence.
A third opinion then holds that in truth the second opinion only forbids even a minute amount of water in a case that one intends to launder, however if one does not intend to launder, such as when using it to filter, then even according to them it is allowed.
Thus according to two of the three opinions mentioned there it is permitted to filter through a cloth when the cloth is placed in ones mouth and the like.
 Shabbos Kehalacha Vol. 2 p. 364
 Ketzos Hashulchan 125 footnote 30, and Shabbos Kehalacha Vol. 2 page 350
 In such a case it has the same status as does a filter, being that once it is designated for that purpose we no longer suspect of the laundering prohibition, as explained in the Alter Rebbe’s reasoning behind why a filter does not contain the laundering prohibition with clear liquids.
 Ketzos Hashulchan 125/15
 So rules Tzemach Tzedek .
 Shabbos Kehalacha Vol. 2 p. 364
 Chayeh Adam, Ketzos Hashulchan, Kitzur SHU”A, M”B
 Ketzos Hashulchan 125/15
 Shabbos Kehalacha Vol. 2 p. 365
 Shabbos Kehalacha Vol. 2 p. 366