Sum of 256

Because all the digits are occurring equal number of times in these numbers at any place (like unit, tens etc) , we can conclude that if we write all one below the other,

then each digit will occur $64$ times in vertical column (:P column is vertical always though :P)

And $1+2+3+4 =10$ , hence if you sum them all, for the unit's place, sum of digits will be $64\times 10$ ,

for the ten's place will be $64\times 10\times 10$ ,

similarly for the thousands place and also for hundred's.

Thus the answer is $64\times (10000+1000+100+10) =64 \times 11110 = \color{#D61F06}{\boxed{711040}}$

In these 256 numbers minimum is 1111 and max is 4444......so avg is (1111+4444)/2.....and the sum of these numbers is 256*(1111+4444)/2=711040

I know this is NT, but just to show a CS solution:

total = 0
for a in range(1, 5):
    for b in range(1, 5):
        for c in range(1, 5):
            for d in range(1, 5):
                total += d * 1000 + c * 100 + b * 10 + a
print(total)