We know, by linearity $$ \Delta (u-v) =\Delta u - \Delta v. $$ However, isn't true that $$ \Delta_p (u-v) =\Delta_p u - \Delta_p v. $$ I'd like to know if it is possible to find a formula
\begin{equation} \Delta_{p}(u-v) = \Delta_{p} u + F(\Delta_{p} v) \end{equation} where $F$ is a real function andthe $ \Delta_{p} $ denotes the usual $ p $-Laplace equation \begin{eqnarray} \Delta_{p} u &: = &\texttt{div} \left ( | Du|^{p-2} Du) \right ) \\ & = & |Du|^{p-4} \left \{ | Du|^{2} \Delta u + (p-2) \sum_{i,j=1}^{n} u_{x_i} u_{x_j} u_{x_i x_j} \right \}. \end{eqnarray} In fact, I'd like to conclude something like that in the particular case where $ v(x) = ax+b $ is an affine function.
My calculations: \begin{eqnarray} \Delta_{p} (u-v) &: = &\texttt{div} \left ( | D(u-v)|^{p-2} D(u-v)) \right ) \\ & = & |D(u-v)|^{p-2} \Delta (u -v)+ (p-2) |Du|^{p-4} \Delta_\infty (u-v) \end{eqnarray} Im particular, if $ v $ is an affine function as above. We have $$ \Delta_{p} (u-v) = |Du -a|^{p-2} \Delta u+ (p-2) |Du- a|^{p-4} \sum_{i,j=1}^{n} (u_{x_i}-a_i) (u_{x_j}-a_j) u_{x_i x_j}. $$ Then, the problem is the leader coefficient $a$.
You essentially answered your own questions by mentioning $v(x)=ax+b$. For this function $\Delta_p v = 0$ so the only way for a formula like $$\Delta_{p}(u-v) = \Delta_{p} u + F(\Delta_{p} v) \tag1 $$ to hold is if $\Delta_{p}(u-v) = \Delta_{p} u$ for affine functions $v$. But the latter is false (try $u$ being the fundamental solution $|x|^{(p-n)/(n-1)}$, for example).
Another way to see that (1) is hopeless is to multiply both $u$ and $v$ by a scalar: since the $p$-Laplacian is homogeneous, the term $F(\Delta_{p} v)$ would have homogeneous too, i.e., of the form $c\,\Delta_{p} v$ with constant $c$. Too much to ask for.
There is a series of papers on superposition of (sub/super)-solutions of the p-Laplace equation. A recent one is Superposition of p-superharmonic functions by Brustad, where you can find references to earlier works. There are no such identities there, they establish inequalities for the p-Laplacian of certain linear combinations.