Is one permitted to wet a clean garment? 

The gemora in Zevachim 94b says that soaking a garment in water is akin to laundering and forbidden. We find a dispute amongst the Rishonim as to whether the gemora is referring to a soiled garment or even to a clean one. According to one opinion the gemora is referring to a soiled garment, because what can be the issur to wet a clean garment. The other opinion holds that there is always room for improvement and wetting a clean garment improves it. [1]

We do not have a clear picture as to how the Mechaber rules, because in simon 302:9 he writes that one may not soak a soiled garment, and in simon 334:24 he writes that there is a problem to soak even a clean cloth in water.

Rav Bentzion Abba Shaul [2] ztz”l learns that the Mechaber holds that soaking only applies to a soiled garment, but says to be stringent and not to soak even a clean garment.

The Rama [3] cites both opinions, and the Mishna Berura [4] says that since soaking a garment involves a Torah prohibition, one should be stringent and not wet even a clean garment.

Therefore the halacha is that one may not wet a clean garment with water. 
 

Is one permitted to make a compress on Shabbos?

A cold compress is often used for treating severe headaches. It is normally prepared by soaking a towel in water, squeezing it out and placing it on the forehead. The squeezing is definitely forbidden. [5] The question refers to the actual wetting of the cloth or towel. One is permitted to wet one's hands and wipe them onto a towel, as we will soon see.

For Sephardim: Rav Bentzion Abba Shaul Ztz”l says that for a sick person it is permitted to soak a clean towel to make a compress. Since this involves a serious offence a Rav should be consulted. [6] The optimal advice would be to have a non-Jew to make the compress. [7]

How is it that I am allowed to dry my hands on a towel on Shabbos, am I not wetting the towel?

This based on another concept called “derch lichluch”, which means that the water is applied in a dirtying manner. When drying hands on a towel, the towel does not become cleaner; on the contrary, it gets dirty. So even though water is coming in contact with the towel, since it is not cleaning the towel, it is permitted. The same rule applies to wiping up spilled water on the floor: even though the water is being absorbed into the towel, since it is done in a “dirtying manner”, it is permitted.[8]

Drying wet dishes with a towel also follows this rule.

If a garment catches fire, may water be poured onto the section that has not yet caught fire?

Of course we are not referring to life threatening cases where anything must be done to save lives. We are referring to a case where one has the time and the cool to concentrate on what he is doing. The Mechaber says [9] that if a garment or cloth caught on fire, one is permitted to pour liquid onto the other side of the garment, thereby causing the fire to extinguish when it reaches the liquid.

All liquids, except for water and white wine soil the garment, and may be poured onto the garment. As for water, it will depend on the opinions mentioned above if one may pour water onto a clean garment. The Bi’ur Halacha [10] says that one should be stringent.