If the product of something is 0, it must be multiplied by 0. Therefore, one of the five integers must be 0.

That gives the following possibilities for 5 consecutive integers:
0, 1, 2, 3, 4
-1, 0, 1, 2, 3
-2, -1, 0, 1, 2
-3, -2, -1, 0, 1
-4, -3, -2, -1, 0

The one with the largest sum will be the one with all positive numbers, so the first one.
0+1+2+3+4 = 10

Therefore, the correct answer is B.