Why can't the Earth's core melt the whole planet? 
Earth core temperature is range between 4,400° Celsius (7,952°
Fahrenheit) to about 6,000° Celsius (10,800° Fahrenheit).

Source
Why can't the Earth's core melt the whole planet? In other words, what is stopping Earth from being melted up to its surface?
 A: Think about a frozen-over lake in the winter. The water underneath is liquid, but it doesn't melt the ice. In fact, it wasn't even able to stop the ice from freezing as the weather got colder in the winter. The surface of the lake was losing heat faster than it could soak up heat from the warmer water below, so it froze while the deeper water was still liquid.
The earth was completely molten right after the impact that formed the moon. That's like the lake at the end of fall. The liquid surface radiated heat away into space until first the surface solidified (pretty quickly) and then the depth of solid rock got greater and greater. The hotter molten rock down below just couldn't heat up the surface fast enough to keep it molten.
A: First thing to notice is that the heat flow is limited, so the heat from the core does not flow to the surface instantaneously.
Second point is that the surface of the Earth radiates energy to the space.
The combination of these effects makes it possible to have a molten core but a cold surface.
A: Melted rock radiates energy, the higher the temperature, the more intense the radiation. In a stable state, the melted rock receives the same amount of energy per second as it gives off by radiation. If it receives less, it will cool down to a lower temperature where the new balance gets established. This lower temperature may be under the melting point of the rock.
Earth's surface rocks are not melted so apparently the heat energy the surface receives from the hot melted rocks inside the Earth per second is small enough to be radiated out by the surface at low temperature, where it is not melted anymore.
A: It's all about temperature gradient and heat current. Think of a rod one of whose edges is maintained at $100^o$C and other at $0^o$C the temperature of a given section would be same and a linear function of distance from either of the end.
Similarly in case of sphere with hot core , the temperature keeps on decreasing as you move away from core and also  the cross-section through which the heat passes keeps on increasing .
The regions where temperature and pressure are right the medium indeed is molten and convection is  the chief mode of heat transfer.
The region surrounding core is theorized to be molten .
A: Most of the source of "heat" in the earth is caused by radioactive decay, apart from what is left over from the original accretion event. Solid rock is a pretty good insulator, so a lot of that heat only reaches the surface slowly.
The inner core is solid iron and the outer core is liquid iron-nickel. the rest of the earth has a composition not unlike the volcanic rock we see, basalt on the ocean floor and more granite-like on the continents (meaning silicon-oxide mostly with mg,al,fe, etc scattered around).
Rock is no different from any other matter, in that it has different phases depending on its pressure, volume and temperature regime. So basically the PVT regime for rock results in most of the earth being solid.
Vulcanism in most cases is the result of tectonic plate motion which can cause friction but also just move rock from one PVT environment to another where it becomes molten and produces vulcanism. There are a few cases of mid-oceanic hotspots, different from plate boundaries, that also produce vulcanism (like Hawaii).
The internal temperature is largely in a steady state with any loss in primordial heat being replaced by radiogenic heat. At the surface of the earth we also have largely been in a steady state with all the incoming radiation from the sun being shed off into space for no net gain. It fluctuates over large periods of time for several reasons, but never so much as to lead to an extinction of all life since the Cambrian. The maximum fluctuation in average surface temperature has been about 12 deg C over the last 500 million years.
A: 
Why can't the Earth's core melt the whole planet? In other words, what is stopping Earth from being melted up to its surface?

I'll rhetorically ask the reverse question: Why can't radiation to empty space freeze the whole planet? In other words, what is stopping Earth from being solid all the way down to its center?
The answer to this reversed question is that that is exactly what is happening, but doing so takes a long, long time, about 4.5 billion years and counting. The Earth's crust apparently formed fairly early on and cooled rapidly, possibly rapidly enough to have enabled liquid water only a few hundred million years after the Earth formed. This is the Cool Early Earth hypothesis.
This ever-present cooling has barely reached the center of the Earth, for several reasons:

*

*The formation of the Earth from many collisions, the differentiation of the Earth into a dense core and rocky mantle and crust, and the hypothesized giant collision with a Mars-sized object made the early Earth have a lot of thermal energy that is still be radiated into space.

*The four key long-lived radioactive isotopes (uranium 235, potassium 40, uranium 238, and thorium 232, listed by increasing half-life) are concentrated in the crust, less concentrated in the mantle, and are probably highly depleted in the Earth's core. Think of these long-lived radioactive isotopes as an electric blanket that keeps the core from losing heat.

*The Earth's mantle and crust are almost 3000 km thick, and rock has a rather low  thermal conductivity compared to other solids. Think of this as a very thick blanket that keeps the core from losing heat.

*Later on, the formation of the inner core has added even more thermal energy to the core. Freezing is an exothermal reaction. This combats the heat transfer across the core-mantle boundary.


Why can't the Earth's core melt the whole planet?

Because the heat flow rate from the Earth's interior is far too small.
Except for a few isolated spots such as geysers in Yellowstone, hot springs in Iceland, and vents near oceanic ridges and underwater volcanos, the heat flow from the Earth is miniscule. Averaged over the surface of the Earth, the heat flow from the interior of the Earth to the surface of the Earth is a bit more than 1/5000 of the heat flow from the Sun and the atmosphere to the surface of the Earth. The Earth's internal energy budget is a noise-level contributor to the temperature of the surface of the Earth.
A: It is worth mentioning that if you double the radius, the surface of a sphere increases by a factor of 4. And the volume of the outer 3000km of the planet is 7 times the volume of the inner 3000km. So there is a lot of opportunity to dissipate heat.
Also, this phenomenon can be observed during eruptions, where lava quickly develops a black crust on the outside, even though it is red-hot just below that small cooler crust.
A: In this answer, I am going to approach the question by approximating the Earth as a uniform spherical ball which has some initial temperature distribution & it is put to some medium with fixed temperature. It does not take into account a lot of things which we might want to take into account when dealing with Earth, but might serve as a good starting point for further calculations. Main point is in the Conclusion section, rest is supporting math.

The equation governing temperature distribution
$$\dot{u}=\alpha \nabla^2 u$$
Lets put this into spherical polar coordinates:
$$\dot{u} = \alpha \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2\frac{\partial u}{\partial r}\right)$$
(We care only about $r$ dependence.)

Boundary and Initial Conditions
The initial temperature distribution is our initial condition:
$$u(r \leq R, t=0) = f(r)$$
The boundary condition is the temperature outside the body (ie sea around polar bear):
$$u(r=R,t \geq 0) = f(R)$$
I am not fond of non-zero boundary conditions. Lets change $u$ to $v$ in a way which make the BCs for $v$ zero:
$$v(r,t)=u(r,t)-f(r=R)$$
Then, $v$ will satisfy:
$$v(R, t \geq 0) = u(R, t \geq 0)-f(R) = 0$$
We solve for $v$ and leater transform back to $u$.
$$v(r \leq R,t = 0) = f(r)-f(r=R) = g(r)$$
$$v(r=R, t \geq 0) = 0$$

Separation of Variables
Let $u(r,t) = \rho(r)T(t)$. Separated equations we arrive at:

*

*Temporal:
$$T'=-\lambda \alpha T$$

*Spatial:
$$r\rho''+2\rho'+\lambda r \rho = 0$$
$\lambda$ is a constant we don't know yet but we count on BCs to get it.


Temporal ODE
Just observe $T$ to be:
$T(t)=Ae^{-\lambda \alpha t}$ for arbitrary $A$.
We'll deal with arbirary constant mulitplicative factors in the spatial solution, so let's just drop the $A$:
$$T(t)=e^{-\lambda \alpha t}$$

Spatial ODE
We are solving $r\rho''+2\rho'+\lambda r \rho = 0$. Lets do series solution, let $\rho(r) = \sum_0^{\infty}c_nr^n$.
We don't want negative powers of $r$ because we want to have a finite $\rho$ at $r=0$.
Subsitute in, compare coefficients, stay physical.
We arrive at:
$$\rho(r)=\frac{c_0}{\sqrt{\lambda}r}\sin(\sqrt{\lambda}r)$$
Our new wonderful BC tells us that $v(R,t \geq 0) = 0$.
Since the temporal part is not going to do it (it is an exponential, not even dependent on $r$), the spatial part has to make it $0$.
When is $\rho(r=R)$ $0$? When the $\sin$ part is $0$. This is the case when $\sqrt{\lambda}R=n\pi$, ie when $$\lambda=\frac{n^2\pi^2}{R^2}$$

Orthonormality is our friend
Rearrange spatial ODE to:
$$-\rho''-\frac{2}{r}\rho' = \lambda \rho$$
LHS is not in Sturm-Liouville form. The weight function which can make it SL type is $w=r^2$.
The eigenfunctions of the SL operator are still:
$$\rho_n(r)=\frac{c_0}{\sqrt{\lambda_n}r}\sin(\sqrt{\lambda_n}r)$$
These are now orthogonal. To make them orthonormal, we need to find $c_0$. What we want is:
$$\int_0^R \rho_n(r)\rho_m(r) w(r) dr = \delta_{nm}$$
Are we integrating over all space or over the all possible values of $r$?
The latter, that's why we don't include an additional $r^2\sin(\theta)$ term in the integrand.
Think about it deeply, why we don't integrate over all space within planet/droplet/polarbear?
We arrive at:
$$c_0  = \frac{n\pi}{R}\sqrt{\frac{2}{R}}$$

How much we want from each eigenfunction
Note that all of the eigenfunctions are solutions to the spatial ODE.
We need to figure out how much we want from each eigenfunction.
All the eigenfunctions are nonzero at $r=0$, and they are all $0$ at $R$;
there are no other eigenfunctions
(we don't use $\cos$ instead of $\sin$ in them, even though they are solutions too, because we want to stay finite at $r=0$).
So the $\rho_n$ eigenfunctions better span the whole function-space. With a bit of notation abuse, we can write:
$$g=\sum_{n=1}^{\infty}\langle g | \rho_n \rangle_w | \rho_n \rangle$$
So the amount of we need from each $\rho_n$ is $\langle g | \rho_n \rangle_w$, ie $\int_0^R g(\zeta) \rho_n(\zeta) \zeta^2 d \zeta$.

Finish
Combine spatial result with temporal dependence, transform back from $v$ to $u$:
$$u(r,t)=\sum_0^{\infty}\int_0^R g(\zeta) \rho_n(\zeta) \zeta^2 d \zeta \rho_n(r) e^{-\frac{n^2\pi^2}{R^2}\alpha t}+f(R)$$

Numerical Results
Absolutely not numpy optimized code can be found here. Temperature evolution with an arbitrarily chosen (not physical) initial temperature distribution, produced by the above code can be observed in this YouTube video.

Conclusion
We can see (especially on the video, or from the form of our solution: note the exponential time decay terms) that boundary conditions are overwhelming the initial conditions over time. This is our main issue: the core is not hot enough to melt the surface given how much heat leaves the surface to space.

This post is based on an earlier writing of mine.
A: Here's a simple answer: All that heat in the core is already being used to keep the core molten. There's not enough heat left to also melt the surface. If their were, the core would have to be hotter.
A: David Hammen has a nice answer, I just would like to add a few interesting notes.
You are asking "Why can't the Earth's core melt the whole planet?", and the answer is that it is trying.
Now let me ask you another question, why can't the extreme cold of space freeze the whole planet through to the core? And the answer is again that it is trying.
The two processes come to a equilibrium and we get a habitable planet with a thin, solid layer that we are actually sitting on.

Please note that contrary to popular belief, the inner core of the Earth is actually solid, and only the outer core is liquid.

Earth's inner core is the innermost geologic layer of the planet Earth. It is primarily a solid ball with a radius of about 1,220 km (760 mi), which is about 20% of Earth's radius or 70% of the Moon's radius.[1][2]

https://en.wikipedia.org/wiki/Earth%27s_inner_core
So we happen to live in an era of Earth with this equilibrium that gives us the habitable top solid layer.

The Sun will exit the main sequence in approximately 5 billion years and start to turn into a red giant.[27][28] As a red giant, the Sun will grow so large that it will engulf Mercury, Venus, and probably Earth.28

https://en.wikipedia.org/wiki/Red_giant
Now if the Sun would not heat our planet then the ultimate fate of the Earth would probably be to freeze to the core in the far future. But we happen to have the Sun and it is heating the planet and in the far future it is said to expand and in that era, it will probably happen as you say that our planet have a molten outer layer (and probably be burned).
