Will all thermodynamic equations have a linear dependence with moles?
Extensive properties actually have a linear dependence with the common magnitudes used for amount of substance.

Mathematical speaking, you work in some representation in which one magnitude is represented with a function of some variables, and the later represent other physical variables. For example $S(U,V,N)$. When a magnitude represents a extensive property, the function is a homogeneous function of of order 1 of the extensive variables, that is:
$S(\lambda E, \lambda V, \lambda N) =\lambda \,S(E,V,N)$

We choose $\lambda = 1/N$, so 
$S(\lambda E, \lambda V, \lambda N) =\lambda \,S(E,V,N)$, using lower case symbols for (say molar) properties,

$ N \,S(e, v, 1) = S(E,V,N)$

So, taking in mind that your equations are formulated for finding values for your thermodynamic functions, they must be stated in the form (for the previous example):

$S(E,V,N) =$ $whatever$

If you work with molar properties you will find just:

$S(e, v, 1) =$ $whatever$ $for$ $a$ $mol$, but using the third equation:

$S(E,V,N) = N\times$ $whatever$ $for$ $a$ $mol$

So the homogeneous (order 1) property will assure you that there is linear dependence with $N$.

Note: Thermodynamic functions that also depend on intensive variables have a zero order dependence in the homogeneous sense with these variables.

EDITED:

After reading the @Curt F. answer, to avoid confusions I decided made a comment: The explaining above works in the scope of 'traditional' thermodynamics, where no surface effects are involved. There are another reasons that can leads to draw wrong conclusions about this linearity, for example, if external fields are involved. But these cases are not in the scope of any course on thermodynamics that I aware of.