Dynamic then should be an outer wrapper not the inner:
list={
{2000,1,3},{2000,1,4},{2000,1,5},{2000,1,6},{2000,1,7},
{2000,1,10},{2000,1,11},{2000,1,12},{2000,1,13},{2000,1,14}};
Method 1 - indexed variable
Column[{
Slider[Dynamic[m],{1,10,1}],
Dynamic[DateString[DateObject[list[[m]]]]]}]
Method 2 - list of variables
Column[{
Slider[Dynamic[m],{DateString[DateObject[#]]&/@list}],
Dynamic[m]}]
Method 3 - localized variable
Manipulate[DateString[DateObject[m]], {m, list, Slider}]
