Basic question about an expression for Gibbs free energy
In the equation 

$$G = \sum \mu_i N_i$$

$G$ is the absolute value of the Gibbs free energy at some point in state space. While not explicitly stated, the values of the chemical potentials $\mu_i$ are in general functions of composition and other state variables (e.g. T and p), rendering $G$ a function of these same state variables.

On the other hand, the equation

$$dG = -SdT + VdP + \sum \mu_i dN_i$$

describes an infinitesimal difference in the free energy between two (infinitesimally close) points in that space.

Neither equation puts constraints on T and p, that is, both are general.