How fast can a "heat front" propagate through a material? Say I have some piece of material (geometry at least I don't think should matter) that is at a uniform temperature. It then is heated (say, by an incident laser beam or something) at one spot, with a uniform rate of heat input. From my understanding of heat conduction, from that moment on, the heat should spread out into the material, with the heated zone advancing steadily. Is there any simple way to calculate, from material properties, how fast the front of that heated zone should be moving?
 A: Yes, this is possible. The core dynamics of this are governed by the heat equation, which for your purposes takes in two key parameters:


*

*the rate of heat input, together with its distribution

*the thermal diffusivity of the material.


Generally speaking, heat doesn't really tend to form "fronts" with a sharply defined boundary. Instead, it acts much more like a diffusing pollutant, with a broad and smooth distribution that moves rather slowly (where 'slowly' means, specifically, that any given point in the distribution will often move with a time dependence of $\sqrt{t}$ instead of at a linear velocity).
The heat equation is a partial differential equation, so it's typically harder to solve than your garden-variety ODE, but as PDEs go it is one of the easiest to solve and one of the most stable to handle numerically. As such, it can be implemented quite easily in a numerical solver, and it can be handled via e.g. Fourier-transform methods, depending on what the geometry is.
A: While Emilio Pisanty's answer gives very valuable information, I feel it doesn't fully answer the title question. Hence my little addition. The heat equation he mentions has solutions that have an infinite speed of propagation, which is a characteristic of parabolic PDEs (which the common heat equation is). And clearly this is not the real speed at which heat propagates.
You mention a laser pointing onto a single material. If that material is a metal, then it is quite common to deal with more complicated PDE than the common heat equation, because the latter one is not a good approximation to describe the temperature anymore. A laser can induce an insane amount of heat in a very small region, causing temperature gradients so high that the temperature variations are of the order of $10^9  \mathrm{K}/ \mathrm{s}$. See this reference for instance.
The speed (as in units of distance divided by time) of heat propagation in a metal is roughly the Fermi velocity (i.e. about $10^6 \mathrm{m}/\mathrm{s}$ for common metals), because that's the speed of the charge and dominant heat carriers (electrons). The case is very different for an insulator or a semiconductor, depending on the doping level. In an insulator the heat is propagated mostly thanks to phonons, whose speed are about 3 orders of magnitude lesser than the speed of electrons, so about $10^3 \mathrm{m}/ \mathrm{s}$. For semiconductors the speed of heat propagation should be roughly in between the one of insulators and metals.
I'll edit my answer later and add more information about heat propagation.
Edit: As promised, here is some information in line with Emilio Pisanty's answer (I think). If we take your case of a uniform temperature material as a 1D semi-infinite region and apply a laser pulse, the maximum temperature region is going to diffuse through the material in such a way that it is going to be attenuated with depth (exponentially) and time. The speed at which this maximum temperature region moves is not constant, it gets slower with time, again exponentially. You can easily visualize it if you play with for example this online heat equation solver. I attach a few pictures showing these effects, at times $t=0$, $0.7$ and $1.3$ (almost twice $0.7$).


]
A: If we approximate the laser as a constant point source of heat (more formally, if we say that we have a delta-function heat source $P\delta(\vec{r})$ for laser power $P$), and if we approximate the armor as a 2D infinite sheet (i.e. if the armor is thin, so that the heat penetrates fully through the thickness of the armor and the temperature is constant as a function of depth), then the solution as a function of time and the distance $r$ from the laser is given by:
$$T(r,t)=\frac{P}{C_m\sigma}\frac{1}{4\alpha}\Gamma\left(0,\frac{r^2}{4\alpha t}\right)+T_0$$
where $C_m$ is the specific heat capacity (heat capacity per unit mass, in J/(kg K)) of the material, $\sigma$ is its density (in kg/m$^2$), $\alpha$ is the thermal diffusivity, and $T_0$ is the initial temperature of the material. The function $\Gamma$ is the incomplete gamma function, and is here equal to
$$\Gamma\left(0,\frac{r^2}{4\alpha t}\right)=\int_{\frac{r^2}{4\alpha t}}^\infty \frac{e^{-z}}{z}dz$$
Unfortunately, this function cannot be written any simpler using elementary functions. We can, however, plot this solution as a function of position and time:

Now suppose that you're interested in the radius $R$ which is hotter than $T_0$ by a specific amount $\Delta T$. This means solving the equation
$$\Delta T = \frac{P}{C_m\sigma}\frac{1}{4\alpha}\Gamma\left(0,\frac{R^2}{4\alpha t}\right)$$
This equation has a partially-analytical solution: using the dimensionless parameter $\eta = \frac{4C_m\sigma\alpha\Delta T}{P}$, we have that
$$R(t)=f(\eta)\sqrt{\alpha t}$$
Once again, there is no analytical expression (that I know of) for $f(\eta)$, but it can be plotted:

For large values of $\eta$ (corresponding to low laser power, high $\Delta T$, high thermal diffusivity, high density, and/or high specific heat), the following approximation appears to hold (I have no idea why this is true, but it works remarkably well asymptotically):
$$f(\eta)\approx \frac{\pi^4}{65}e^{-\frac{\eta}{2}}$$ 
