Is Fourier's law of conduction a consequence of the second principle? In classical thermodynamics courses, entropy is often motivated by the need to justify that heat flows from high temperatures zones to lower temperatures zones: this is seen as a consequence of maximizing entropy. However it could also be seen to be a consequence of the law of heat conduction. Looking up in the non-equuilibrium thermodynqmics literature I can see a connection with Onsager's reciprocal relations but from what I can tell they are introduced as phenomenological laws.
Can the law of conduction be derived from the assumptions of classical thermodynamics?
 A: When we speak about heat conduction we must use the laws of continuum thermomechanics, where quantities like temperature, internal energy, etc. can vary from place to place.
The second law of thermodynamics for a body $B$ when the body is at rest and the only form of heating is by contact (as opposed to bulk heating, as in a microwave oven for example) can be written this way:
$$\frac{\mathrm{d}}{\mathrm{d}t}\int_B s \,\mathrm{d}v \geqslant - \int_{\partial B} \frac{\boldsymbol{q}}{T} \cdot \mathrm{d}\boldsymbol{a}\tag{*}\label{2ndlaw}$$
where $\partial B$ denotes the surface of the body, $s$ is the entropy per volume, $\boldsymbol{q}$ is the heat outflux (energy/time/area), $\mathrm{d}v$ is the volume element and $\mathrm{d}\boldsymbol{a}$ the area element at the surface, with outward-pointing normal. Note that $\boldsymbol{q}\cdot\mathrm{d}\boldsymbol{a} < 0$ means that the body is heated. The inequality above is the "Clausius-Duhem inequality" mentioned in hyportnex's answer.
If the temperature always stays uniform throughout the body, the inequality above reduces to $\mathrm{d}S/\mathrm{d}t \geqslant Q/T$, typical for uniform processes, where $S$ is the total entropy of the body and $Q = -\int_{\partial B} \boldsymbol{q}\cdot\mathrm{d}\boldsymbol{a}$ the total heating (energy/time) of the body through its surface.
We can rewrite the inequality $\eqref{2ndlaw}$, which also holds for any part of the body, in local form using Gauss's theorem:
$$\frac{\partial s}{\partial t} \geqslant -\nabla\cdot\frac{\boldsymbol{q}}{T}
\equiv -\frac{1}{T}\nabla\cdot \boldsymbol{q}
+ \frac{1}{T^2}\boldsymbol{q} \cdot\nabla T\tag{**}\label{2ndlawloc}$$
(again, this is a special form valid when the body is at rest and the only form of heating is by contact).
In steady-state conditions the entropy and internal energy don't change with time, so the term on the left side and the first term on the right side (which by the first law equals the increase in internal energy divided by $T$) vanish. Considering that absolute temperature is positive, we are left with
$$\boldsymbol{q} \cdot \nabla T \leqslant 0,\tag{***}\label{heatcond}$$
which says that the heat flux must form an obtuse angle with the temperature gradient – in other words, "heat flows from hot to cold", which I suppose is what the question refers to as "the law of conduction". So: yes, this law can be derived from the second and first laws for continua – under a steady-state condition in a rigid body at least, for example in an iron bar which has been heated at one end and cooled at the other at a constant rate for some time. 
But in more general situations the inequality $\eqref{heatcond}$ needs not hold.
That it needs not hold in general is clear from the local form $\eqref{2ndlawloc}$. For example, let me quote from Astarita (1990), § 7.1: 

the second law reduces to the requirement in equation $\eqref{heatcond}$ only for steady-state phenomena. In other words, for unsteady-state phenomena, the second law does not forbid heat to flow in the direction of increasing temperature, if only for short intervals of time.

He continues in § 7.5:

Even if the isotropic form of Fourier's law has been established experimentally for steady-state conditions, it need not hold also under unsteady-state conditions. Indeed, consider the following constitutive equation for the heat flux [...]
  $$\boldsymbol{q} + \theta \frac{\partial\boldsymbol{q}}{\partial t} = -k \nabla T\tag{7.5.3}\label{astarita}$$
  with $\theta$, the relaxation time for heat flux, a positive constant. This equation is guaranteed to deliver a heat flux vector which, in steady state, forms an obtuse angle with the temperature gradient vector, and its validity (or lack thereof) cannot be ascertained by steady-state experiments.
At unsteady state, equation $\eqref{astarita}$ does allow the heat flux vector to form an acute angle with $\nabla T$, as the following simple example shows. Suppose $\nabla T$ has been held constant at some value, and correspondingly the heat flux forms an obtuse angle with it. At some time $t = 0$, the temperature gradient is suddenly reversed. The heat flux will also reverse in a time scale of order $\theta$, but at $t$ larger than $0$ by an amount negligibly small as compared to $\theta$ it will still have the direction it had at negative times–and hence it will form an acute angle with $\nabla T$.
This, however, does not contradict the second law, since equation $\eqref{heatcond}$ requires the heat flux to form an obtuse angle with $\nabla T$ only at steady state. If a finite relaxation time $\theta$ is allowed for, the usual Maxwell relations do not hold at unsteady state, and hence the other terms appearing in equation $\eqref{2ndlawloc}$ may well compensate for a positive value of the last term on the left-hand side.
Indeed, some experimental results on the rate of crystallization in polymers suggest that a constitutive equation for the heat flux of the type of equation $\eqref{astarita}$ is needed in order to model the data.

What Astarita says also implies that Fourier's law of conduction, $\boldsymbol{q} = -k(V, T) \nabla T$, cannot be derived from the second law only: it is a constitutive equation, that is, an equation specifying the heat-conduction properties of particular bodies only. Other bodies may satisfy different laws (cf. Astarita's $\eqref{astarita}$). Fourier's law $\boldsymbol{q} = -k(V, T) \nabla T$ can be derived by assuming, besides the second law, also a dependence of the fluid properties on particular variables and a form of linearity; see for example Samohýl & Pekař (2014), §§ 3.5–7, and the references there.
For the history and other comments about the second law for continua $\eqref{2ndlaw}$ see Truesdell (1984).
References


*

*Astarita, G (2000): Thermodynamics: An Advanced Textbook for Chemical Engineers (Springer).

*Samohýl, I., Pekař, M. (2014): The Thermodynamics of Linear Fluids and Fluid Mixtures (2nd ed., Springer).

*Truesdell, C. A., editor (1984): Rational Thermodynamics (2nd ed., Springer).
A: Short answer is no but even then it depends on what you mean by "the 2nd law of thermodynamics". In conventional treatments of so-called equilibrium thermodynamics Fourier's law of heat conduction is completely independent of the rest. In what is called "rational thermodynamic" where the 2nd law is formulated as the "Clausius-Duhem inequality" it is in fact becomes part of the "2nd law" and a generalization of it as well. From the Clausius-Duhem inequality it can be shown that for heat conduction in the linear regime the conductivity must be positive or if in an anisotropic crystal a positive definite tensor. The symmetry of the tensor would follow from the so-called Onsager's reciprocity principle but Truesdell claims it has never been verified experimentally for all crystalline classes, but his, C. A. Truesdell: Rational Thermodynamics where you can read quite a lot on this subject is an "old" book, so there maybe newer experimental results on that. In fact, Truesdell uses the paucity of experiments on the symmetry of the heat conduction tensor to denounce "Onsagerism" as a quasi-religious movement that has never produced much of anything. The same formalism is used to introduce the "rational thermodynamics" of diffusion.
A: One cannot "derive" any non-equilibrium rate law from thermodynamics, simply because they are beyond the scope of the theory. Thermodynamics simply does not deal with such phenomena and hence cannot tell you how such processes occur (in this case heat conduction). All that thermodynamics does is relate mean values of certain properties of systems amongst each other (which also implies that second order quantities like specific heat which depend on property fluctuations cannot be derived in thermodynamics and are instead an external input to the theory). Instead, a so called derivation of Fourier's law is still primarily phenomenological and can be done, for which one must have a look at Chaikin and Lubensky's book on Principles of Condensed Matter, under the topic of Hydrodynamics.  
I shall just briefly mention some of the points as given in the book. The Fourier law can be thought as the lowest order contribution in a gradient expansion, thereby considering only small thermal gradients. There is no reason for linear response theory to be valid in a large temperature range, if it is, then it is a peculiarity of the material. Considering the slow spatial modulation in temperature as a long wavelength thermal excitation and the conservation of energy, we can systematically write down the linearized equation for the energy current, i.e heat. This would directly give you Fourier's law in the process. The associated transport coefficient is labelled the thermal conductivity and whose sign is fixed by the second law of thermodynamics, for stability reasons.  
A detailed procedure is provided in the book mentioned above.
A: 
Can the law of conduction be derived from the assumptions of classical thermodynamics?

The answer should be no, because classical thermodynamics does not deal with description of irreversible processes in time; it only deals with equilibrium states. Second law of thermodynamics does not assert that entropy decreases in time, only that after irreversible process ends up in a new equilibrium state, entropy could not have decreased.
The Fourier law of heat conduction is a way to describe what happens to temperature of an object in time. In one way, it is more general description, because it describes non-equilibrium state, but in other way it is also less general, because it does not apply to all heat conduction processes.
It seems strange, but these two theories of heat are largely separate - they deal with different kind of questions about heat and temperature. Perhaps this means we do not have satisfactory theory of them (irreversible thermodynamics) yet.
