Two significant and profound examples concern the theory of superconductivity and the the thermoelectric effect.
Example one: electromagnetic response of a superconductor
Consider the frequency and wavevector dependence of the current-current correlation function, denoted Lambda(q,omega) below, where q is the wavevector and omega the frequency. The q to 0 limit can also be viewed as taking the thermodynamic limit.
The following equations are taken from an important 1993 paper Insulator, metal, or superconductor: the criteria by Doug Scalapino, Steve White, and Shoucheng Zhang
The first equation defines the superfluid density D_s, and the last one the Drude weight D. The two quantities are equal in a BCS superconductor but are not equal in a non-superconducting metal or in unconventional superconductors such as cuprates or organic charge transfer salts.
The second equation is required by gauge invariance. Comparing to the first equation we see that one gets a different answer depending on the order in which q_x (parallel to the current) and q_y goes to zero.
Example two: thermoelectric response
A very elegant 2010 paper Kelvin formula for thermopower, Michael Peterson and Sriram Shastry considered the significance of taking limits in different orders. They started with the exact Kubo expression for the frequency and wavevector dependent thermoelectric response S(q,omega).
The correct value for the Seebeck coefficient is obtained by first taking the q to 0 (i.e. thermodynamic limit of infinite system size) and then taking the static (omega to 0) limit. However, if one reverses the order of these two limits S(q,omega) reduces to Kelvin's (1854) formula which gives the Seebeck coefficient (n.b. a transport property) in terms of purely thermodynamical variables, i.e. S is proportional to the derivative of the chemical potential with respect to temperature. This formula is only approximate, but is very useful particularly for getting magnitudes and trends.
The slide below from a 2012 talk by Shastry nicely summarises the above.