jmc

Here is a sequence of consecutive integers that add up to $2014$: $$-2013,-2012,\dots ,-1,0,1,\dots ,2012,2013,2014.$$So $-2013$ is yummy.

Assume there is a yummy integer less than $-2013$. Then there is a sequence of consecutive integers (including at least one less than $-2013$) that add up to $2014$. Let $A$ be the least integer in the sequence, so $A<-2013$.

Because the sum of the sequence is nonnegative, it includes the numbers $A,\dots ,-1,0,1,\dots ,-A$. Because the sum of the sequence is positive, besides the numbers above, it includes $-A+1$. But $-A+1>2013+1=2014.$

So the sum of the sequence exceeds $2014$, which is a contradiction. Hence there is no yummy integer less than $-2013$.

Therefore the least yummy integer is $\boxed{-2013}$.