It's quite simple to see that no such identity holds, which breaks my heart. I'll try to find some pieces of it and put it back together at the end of this answer.
Ok, ONE trivial identity holds , which is pretty obvious : if you sum up every kind of expression of the form $(\pm n_1 \pm n_2 \pm n_3)^3$ by taking all $\pm$ combinations, you'll get zero, simply because $(-a)^3 = -a^3$ for all $a$ and every expression above can be paired to its negative counterpart in the summation.
In fact , because of the pairing, we can restrict the possible $\pm$ combinations to $4$ instead of $8$, because half of them are a negative of the other.
But apart from that, we can't have an identity of the form :
a(n_1^3+n_2^3+n_3^3) = a_0(n_1+ n_2+ n_3)^3 + a_1(-n_1+n_2+n_3)^3 + a_2(n_1-n_2+n_3)^3 + a_3(n_1+n_2-n_3)^3
for any $a \neq 0$. Let's prove it by contradiction : we proceed to assume that this equality holds, and then demonstrate another equality from here which is clearly false.
First, match the coefficients of the $n_i^3$ together. From the multinomial expansions of the sums of cubes, we know that the coefficient of $n_i^3$ on the right hand side is $(a_0+a_1+a_2+a_3) - 2a_i$ (which is basically saying that you subtract the $a_i$ corresponding to the $n_i$, but the rest get added). But all the $n_i^3$ have coefficient $a$ on the LHS. This reasoning, along with simple algebra, leads to the conclusion that $a_1=a_2=a_3$ and $a = a_0+a_1$.
Thus, we are left with the identity :
(a_0+a_1)(n_1^3+n_2^3+n_3^3) = a_0(n_1+n_2+n_3)^3 + a_1[(-n_1+n_2+n_3)^3 + (n_1-n_2+n_3)^3 + (n_1+n_2-n_3)^3]
Transpose to get :
(a_0+a_1)(n_1^3+n_2^3+n_3^3) - a_0(n_1+n_2+n_3)^3 = a_1[(-n_1+n_2+n_3)^3 + (n_1-n_2+n_3)^3 + (n_1+n_2-n_3)^3]
Let's compare the coefficients of $n_1n_2n_3$ on both sides now. On the LHS, it is seen to be $-6a_0$. On the RHS, it is found to be $-18a_1$. (All this can be seen from the multinomial expansion). It follows that $a_0 = 3a_1$ and therefore $a = 4a_1$.
All this clearly proves that if $a_1 = 0$ then the identity has $0$ on both sides, hence makes no sense. If $a_1 \neq 0$ then we can cancel out $a_1$ from both sides of that equation, and we get :
4(n_1^3+n_2^3+n_3^3) - 3(n_1+n_2+n_3)^3 = (-n_1+n_2+n_3)^3 + (n_1-n_2+n_3)^3 + (n_1+n_2-n_3)^3
So if any identity of the kind we want holds true, it MUST be this identity. Proving that this identity isn't true is fairly clear : we just find one triple $n_1,n_2,n_3$ for which it's false. I took the most obvious triple that came to my mind (note that I couldn't take all of $n_1,n_2,n_3$ equal!) So I took $n_1=n_2 = 2$ and $n_3 = 1$ and I got the LHS as $-307$, while the RHS came out as as something much bigger.
Thus we complete the contradiction : and no such identity exists.
For comparison, note that the identical identity with squares does work out , but things over there are slightly different. First of all ,we have $(-a)^2 = a^2$ so while opposing terms (in $(\pm x \pm y)^2$) don't cancel out, they do add up which means any identity is going to look like :
a(x+y)^2 + b(x-y)^2 = c(x^2+y^2)
Comparing the coefficients of $x^2$ or $y^2$ tells you that $a+b = c$ ,and comparing the coefficients of $xy$ tells you that $a=b$. So once again, dividing by $a$ tells you that the only identity that can be true is $(x+y)^2+(x-y)^2 = 2(x^2+y^2)$, which turns out to be true, instead of false.
There are, however, many identities regarding sum of cubes : these don't have arbitrary bases, though. Here is a minefield of algebraic identities, not just about cubes but about general polynomials of varying degrees in many variables. This is enough to get you through a life of boredom and broken hearts.
Another interesting question that comes out from the question you proposed , is linked with sums of squares : the very interesting identity $2(m^2+n^2) = (m+n)^2+(m-n)^2$ actually brings the following interpretation :
If $K$ is a sum of two integer squares, then $2K$ is also a sum of two integer squares.
Indeed, this interpretation hasn't been afforded to cubes yet, as far as I know. I believe that representation as a sum of three cubes hasn't been characterised, while representation as a sum of two cubes and four or more cubes are certainly better known. Following a perusal of these results, I had come to the conclusion that no identities such as the one used in the squared case were available there, and so the kind of identity proposed wasn't going to work out.