Almost, but generally no1
There are no such metal, alloy or pure element which does this within a reasonable temperature-pressure range2; but there are materials which can behave this way.
Why does stuff generally melt when heated and not when cooled down?3
All matter in our universe follow the bureaucracy founded within thermodynamics and, as part of that, all systems tries to reduce their internal energy to a minimum. A way to express the usable energy in a system is through Gibb's free energy, which tells how much energy can be extracted from a system. Gibb's free energy is commonly expressed as:
$\Delta G = \Delta H - T\Delta S$
in which $\Delta G $ is the Gibb's free energy, $\Delta H$ is the enthalpy (energy of the system, including the internal energy), $T$ is the absolute temperature, and $\Delta S$ is the entropy (the "disorder" of the system). If $\Delta G $ is negative, then it is a spontaneous process.
When atoms and molecules becomes highly organized, then they can usually lower their energy by a fairly decent amount, but they lose "disorder". This means that both $\Delta H$ and $\Delta S$ are negative for something which solidifies, but as $\Delta S$ is preceded a minus sign in the equation, a loss of disorder means that it adds to $\Delta G$. Luckily, it is multiplied by $T$, so it contributes less to the equation at low temperatures. The more negative $\Delta H$ can become for a system which is solidifying, the higher $T$ it can allow without $\Delta S$ taking overhand. As long as the sum is negative, then it will spontaneously solidify.
Similarly, when you increase the temperature you start to add energy to the system in terms of heat and the molecules start to move around. This creates a positive $\Delta H$, but at the melting point they start to gain enough disorder ($\Delta S$ is now positive too) from the movement so that $T\Delta S$ becomes larger than $\Delta H$. Since the "disorder" gain contributes negatively, this causes $\Delta G$ to become negative again and the system will spontaneously melt and the "disorder" in the system will keep it melted for as long as $T$ is high enough.
Things can solidify when heated, but irreversibly
Polymerization is one method of creating solids by heating; it is what the reaction is called when you create plastics. You start of with monomers (the individual building blocks), which react through various means into polymers (i.e., plastics). There are several different ways for the monomers to react (e.g., free radical polymerization) and there are several ways to initiate the polymerization process (e.g., through light and heat). The polymerization process is also not limited to petrochemical plastics; similar processes also occur with biopolymers, such as milk proteins and wheat proteins However, this process is usually one way, meaning that once the polymer is created, then there is no melting of it, so it remains solid.
However, there is a novel material which breaks this rule and, seemingly, defies nature. It is an aqueous mixture of cyclodextrin and 4-Methylpyridine, which solidifies through gelation when heated above 45°C4 and is stable due to hydrogen bond formation between the two compounds. The bonds are created in a certain way as the cyclodextrin changes shape when heated and, when the mixture is cooled, the hydrogen bonds are broken and the cyclodextrin changes back to it's original shape and is, thus, prevented from forming the bonds required to solidify the solution.
So, while this example is not a metal alloy, it still shows that there is a material which can solidify when heated at a reasonable temperature range.
1: I misread the question and thought it said "any material" when writing my first version of my answer.
2: As Cort Ammon pointed out, Helium-3 does actually do this, but in a very small window at extremely low temperatures, which makes it highly impractical to use.
3: I'm trying to keep it simple and I am, thus, taking some liberties with the truth.
4: The article states "between 45°C and 75°C, which I interpret as that the compound is destroyed if heated more.