The most prominent use of fireships in Chinese warfare was during the Battle of Red Cliffs, during the Three Kingdoms period in 208 CE. I'm not actually aware of any other instances of fireships being used in early imperial Chinese warfare, but am more than happy to be proven wrong. (There seems to have been some use of fireships in the Song over a millenia later.)

The fireships were used by the Wu general Huang Gai 黃蓋 as part of a ruse. The 54th fascicle of the Records of the Three Kingdoms (not to be confused with the fictional and MUCH MORE well-known Romance of the Three Kingdoms) records them as such:

瑜部將黃蓋曰：「今寇眾我寡，難與持久。然觀操軍船艦，首尾相接，可燒而走也。」乃取蒙沖鬥艦數十艘，實以薪草，膏油灌其中。...蓋放諸船，同時發火。時風盛猛，悉延燒岸上營落。頃之。煙炎張天，人馬燒溺死者甚眾。

"Yu's subordinate general Huang Gai said: "At present, the enemy is numerous and our [troops] are few; this is a difficult [situation] to maintain for long. However, I have observed that the warships of the Cao military are linked together, front and back. We can burn and clear them out." He then took ten mengchong warships and filled them with kindling and grass soaked with grease and oil... Gai released the boats and they ignited at the same time. At the time, the wind was vigorous and fierce, completely extending to the army encampments on the bank. In a moment, smoke and flames enveloped the sky, and there were numerous men and horses that burned or drowned to death."

The mengchong ships of the period, according to Zhang Chunhui 张春辉 in 中国机械工程发明史 The History of Chinese Inventions in Machinery and Engineering were a type of "troop-carrying, well-armored, and fast-moving transport ship." (333) Hence they were not purpose built for being set alight in warfare, unlike the Hellburners, but rather, fast logistics vessels converted for the purpose. I cannot speak to the influences the Dutch might have gotten their ideas from.