Which of two objects at the same tempreature can cause more intense burns when you touch it: the one with the greater specific heat capacity or the one with the smaller specific heat capacity and why?
Assuming the same mass of the object, the answer is: One with greater specific heat.
As you touch an object heat from the object transfers from the object (warmer object) to your body (colder object). By this process object cools for certain temperature. Greater specific heat means greater heat transferred for the same temperature difference and greater heat/energy means potentially larger injury/burns.
Consider a spoonful of really hot soup, and say a french fry of similiar temperature - expereince tells you that you burn your mouth 'more' on the soup. As stated, the main difference is the the heat capacity of the soup, that's high because of the high water content.
So yes, a higher heat capacity means you can get more severe burns - assuming beeing burnt badly correlates directly with termal energy flow into your hand (or whatever), and the temperature at which this happens. The with a high capacity object loses temperature slower, so there is a higher $\Delta T$ (temperature difference) for longer, meaning more energy transfer.
A second important characteristic (that is easily checked with a cold piece of metal) is thermal conductivity - the better a thermal conductor your object is, the faster heat flows into the area that you cooled by touching it.
(pygmalions answer including the first comment says the same thing)