What is the lower bound to the temperature at which a fire can burn?
Fires which burn below about 400°C are known as cool flames, and this is a well known and widely researched phenomenon. Most hydrocarbons and many alcohols can produce cool flames, and the conditions under which they burn depends on the oxygen content available. The most common encounter that most people have with the cool flame effect is through incomplete combustion of a petrol engine, leading to engine knocking. 

The lowest recorded cool flame temperatures are between 200 and 300°C; the Wikipedia page references n-butyl acetate as 225°C. You can read a lot more about cool flames on that page. One thing to note is that cool flames are very hard to visibly detect at lower temperatures - both heat and light being two of the by products of the combustion process.

NASA have conducted a number of experiments in space on low temperature combustion of nano droplets of heptane, and this is an interesting area of science still being investigated.

An interesting point to make relates to your question of whether you could hold your hand in the flame and not burn yourself. Clearly the answer is no, but for some cool flames it may take noticeably longer to feel the pain of heat. Paper starts to discolour at about 150°C, but may not burn as ignition temperature is between 220-250°C. For reference, candle flames are about 600-1400°C.