If some sort of Najas (i.e. semen and dog excrement, etc.) came onto a cloth that could only be dry-cleaned, would dry-cleaning be sufficient for the removal of this impurity?

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

There is a difference of opinion among the scholars regarding the issue of removing impurity with something other than water. Most of them are of the opinion that the use of water is obligatory to remove all sorts of impurities except what Islam permitted to be purified with other than water, like shoes and the women's clothing that trails on the ground and the use of stones to purify oneself from excrement.

Imaam An-Nawawi (from the Shaafi'i School) said in Al-Majmoo': "Removing impurities and purifying oneself from the state of impurity can not be done except with clean water [i.e. the water whose taste, smell and color did not change]." He added: "This is the opinion of the Shaafi'i School and it is reported from the majority of scholars from among the Salaf (rightly guided predecessors) i.e. the Companions and those who came after them."

However, Abu Haneefah is of the opinion that impurity can be removed with any liquid that is pure. Ibn Taymiyyah's, opinion is close to Abu Haneefah's.

Nonetheless, Ibn Taymiyyah has a more detailed opinion about this matter: He said: "There are three opinions regarding removing impurity with something other than water. The first opinion is that As-Shaafi'i forbids it. The second opinion is the that of Abu Haneefah who considers it to be lawful. This is one of two statements of both Imaam Maalik and Imaam Ahmad . The third opinion is the opinion of the Hanbali school, which is that it is acceptable to remove impurity with something other than water when there is a necessity."

Ibn Taymiyyah then said: "The Prophet's Sunnah orders us to use water in removing impurities for the saying of the Prophet to Asmaa': "Scrape if off then rub it using water, then wash it with water, and then pray on it." So, the Prophet ordered us to use water to remove impurities in some special matters and he did not give a general order to use water in removing all kinds of impurities. He (the Prophet ) also ordered the removal of impurities with things other than water, like removing excrement with stones." 

Then Ibn Taymiyyah said: "If this is the case, then the preponderant opinion regarding this issue is that whenever impurity is removed using any means, its ruling is cancelled (i.e. it becomes pure). In fact when the ruling is attested through a given cause, it is cancelled when that cause is removed.' [Al-Fataawi Al-Kubra (Volume 1), the Book of Purity].

There is no doubt that acting according to the opinion of the majority of the scholars, which is that impurity can only be removed with pure water, is safer, so a person who fears Allaah should opt for it.

Allaah Knows best.