If the heat transfer in a particular direction is negligible then is it necessary that the temperature variation will also be negligible? Imagine having a rectangular block. If the heat transfer is predominant in only one direction (say x) and the heat transfer in the other two directions (say y and z) is negligible, then can I conclude, always, the temperature variation in the y and z directions will be very small?
 A: 
If the heat transfer in a particular direction is negligible then is
it necessary that the temperature variation will also be negligible?

Yes, assuming isotropic material behavior (thermal conductivity the same in all directions).
It follows from Fourier's Law of Heat Conduction that
$$\vec q=-k\nabla T$$
Where
$\vec q$ = heat flux (W/m$^2$)
$k$ = thermal conductivity (W/(m.K)
$\nabla T$ = temperature gradient (K/m)
Then, for an isotropic material
$$\vec q=-k\biggl(\frac{\partial T}{\partial x}\vec i+\frac{\partial T}{\partial y}\vec j+\frac{\partial T}{\partial z}\vec k\biggr )$$
From which it follows that if the heat flux is in the x direction only, then the temperature variation (gradient) in the y and z direction is zero.
Hope this helps.
A: 
If the heat transfer in a particular direction is negligible then is it necessary that the temperature variation will also be negligible?

No.
Heat transfer is mediated by both the temperature gradient and the geometry and material properties, which can vary with direction. If heat transfer in a particular direction is negligible, then either the temperature gradient is very small, or the thermal conductivity in that direction is very small, or both, but I cannot exclude the latter possibility (unless you also guarantee that the thermal properties of the block are isotropic, for example). This is an important aspect of modeling laminates, for instance.
