It is true and well-known (I am assuming that $F$ is locally uniformly elliptic), but I am not sure of a precise reference. Interior smoothness follows from interior Schauder estimates (applied to difference quotients of $u$ and its derivatives, successively). To extend to the boundary, one uses boundary Schauder estimates. One can reduce to the case of zero boundary data and flat boundary (that is, $\Omega = B_1^+$ and $u = 0$ on $\{x_n = 0\}$) after subtracting $\phi$ and performing a diffeomorphism, which don't change the class of equations under consideration. The difference quotient method and boundary Schauder estimates show that $u_i$ are $C^{2,\alpha}$ up to the flat part of the boundary for $i < n$. By the uniform ellipticity of the equation, $u_{nn}$ can be written as a smooth function of $u_{ij}$ for $(i,\,j) \neq (n,\,n)$, $\nabla u$, $u$, and $x$. All of these quantities are $C^{1,\,\alpha}$ up to the flat part of the boundary, hence $u \in C^{3,\,\alpha}$ up to the boundary. Higher regularity follows after differentiating the equation (the coefficients of the differentiated equation are now $C^{1,\alpha}$) and applying a similar procedure. (One in fact only needs $u \in C^2\left(\overline{\Omega}\right)$; the first step uses instead the Calderon-Zygmund $W^{2,\,p}$ estimate for the equation solved by the difference quotients to get $C^{2,\alpha}$ regularity via embeddings (take $p > n$), and then proceeds as before).

It is true and well-known (assuming $F$ is e.g. uniformly elliptic). The idea is sketched in ch. 9 of the book by Caffarelli-Cabre, but I am not sure of a precise reference at this level of generality. Interior smoothness follows from interior Schauder estimates (applied to difference quotients of $u$ and its derivatives, successively). To extend to the boundary, one uses boundary Schauder estimates. One can reduce to the case of zero boundary data and flat boundary (that is, $\Omega = B_1^+$ and $u = 0$ on $\{x_n = 0\}$) after subtracting $\phi$ and performing a diffeomorphism, which don't change the class of equations under consideration. The difference quotient method and boundary Schauder estimates show that $u_i$ are $C^{2,\alpha}$ up to the flat part of the boundary for $i < n$. By the uniform ellipticity of the equation, $u_{nn}$ can be written as a smooth function of $u_{ij}$ for $(i,\,j) \neq (n,\,n)$, $\nabla u$, $u$, and $x$. All of these quantities are $C^{1,\,\alpha}$ up to the flat part of the boundary, hence $u \in C^{3,\,\alpha}$ up to the boundary. Higher regularity follows after differentiating the equation (the coefficients of the differentiated equation are now $C^{1,\alpha}$) and applying a similar procedure. (One in fact only needs $u \in C^2\left(\overline{\Omega}\right)$; the first step uses instead the Calderon-Zygmund $W^{2,\,p}$ estimate for the equation solved by the difference quotients to get $C^{2,\alpha}$ regularity via embeddings (take $p > n$), and then proceeds as before).