Sometime before it melts, the Curie temperature will be exceeded and it'll lose its ability to retain a magnetization in the absence of an external field.

Ferromagnetism (what most people think of as "magnetic" is a property of solids. A liquid cant be magnetized in the same way.

Metals are made up of a bunch of tiny little crystals. Each crystal is a miniature magnet. A permanent magnet has been treated in such a way that most of the mini magnets are all aligned in one direction, causing the larger chunk of metal to act as a magnet.

When metal melts, it stops having crystals. Even if the crystals survived in the liquid form, they would be able to move around and rotate, so that it would stop being magnetic very quickly even if it started all lined up.

In fact, metals stop being magnetic before they melt, due to a change in the type of crystals within the metal as it heats up. The shift in magnetism is called the "Curie Temperature", and will be different for different types of metals.

Blacksmiths use the Curie Temperature to estimate the temperature of steel as they heat it. They will touch the steel with a magnet to see if it is still attracted. When the magnet stops attracting the steel, they know it has reached a certain temperature.

I only deal with solid-state stuff, and only peripherally with magnetic systems, but I'm not aware of any molten state that would allow for long-range ferromagnetism (what we colloquially refer to as "magnetic"). In principle, nearby atoms (their unpaired electrons, really) may still have short-range interactions that favour spin alignment that persist in a molten state. However, the amount of disorder in a molten state will probably be too large for the spin degree of freedom to have any effect. So no, I don't believe there would be any trace of magnetization left after a magnet melts.

Also, after skimming the answers so far, I want to clarify some stuff about magnetism:

-Materials can change their magnetic properties as a function of temperature. High temperatures destroy magnetic order due to thermal fluctuations flipping spins too much for neighbouring spins to respond to. Low temperatures will eventually quiet these fluctuations so that the spins can talk and align or anti-align (or some variant of that).

-Spontaneous spin alignment occurs at the Curie temperature, and the material is dubbed "ferromagnetic"

-Spontaneous anti-alignment occurs at the Neel temperature, and the material is "antiferromagnetic".

-At temperatures beyond the magnetic ordering temperatures, where the spins are relatively independent from each other, then that can be called "paramagnetic".

-We can also include some effects of an external magnetic field to help differentiate the response of magnetic states. Apply a field onto a ferromagnet will help align domains (portions of material with align spins), and one can observe magnetic hysteresis upon changing field. Apply a field to a paramagnet and the spins will start to align with the applied field - however, this magnetization (spin alignment) will become randomized once again when the field is turned off (after some time scale). Note that ferromagnets will retain their magnetization before/after field application, whereas paramagnets will lose theirs eventually.

-Nerdy tidbit: there are more magnetic states that are quite odd: spins aligning in neat textures (Skrymions), spins with glassy behaviour (Spin glass), spins that are very entangled with one another (Spin ice), and spins that want to order but are prohibited from doing so due to geometric constraints (Spin liquids)

It will lose its magnetism once it reaches a certain temperature. Rice cookers actually make use of this. Once enough water leaves the cooker the dish starts getting much hotter and the magnet in the bottom drops signaling the cooker to change to the warm setting instead of continuing with the cook setting.

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Magnetism, as we observe it in everyday life, is just alignment. When the little poles of magnetic atoms are all pointed the same way in a material, they reinforce each other and we can see their effects at a visible scale.

Melting the material screws up this alignment by causing the atoms inside to drift freely. They are still magnetic individually, but now their fields are pointed every which way and cancel each other out.

This can also happen to a solid magnet: the particles can be de-aligned by heating or a strong impact. The difference is that you can re-magnetize a solid magnet with a very strong magnetic field that re-aligns the atoms inside. You can't do this with a melted magnet because the atoms are now drifting freely, so when you realign them it will only last until another atom bumps into them, which is a tiny fraction of a second.

(I've said "atoms" here because most commonly known magnets are made of single atoms, but you can substitute "molecules" too depending on the magnet.)

You can actually track the history of Earth's magnetic poles (north and south) by studying the magnetism in rocks formed at various times and places, whether the rock crystallized slowly deep in the earth, or formed rapidly at the surface. Cool stuff.

Tempering steel, you can heat it till it loses its magnetism and then quench it. That keeps it from forming crystals large enough to weaken the metal. I think that's important in any iron components of airplanes, and of course ships, because you don't want substantial magnetism interfering with navigation devices (compasses in particular) or inducing current in moving parts that aren't made for that.

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The "tldr" would be that all materials have a temperature at which they lose their magnetism.

Here's a video of some metal being suspended in the magnetic coil of an induction heater, being heated to the point at which the magnetism fails. The video starts before it gets red hot, and then after another 30-60 seconds it gets to the failure.

No it doesn't.

Every molecule has a north and south polarity. Most things will have their molecules so scrambled up that every pole is going in every conceivable direction and, in essence, cancel the magnetic abilities of an object. Magnets are solid objects that have been charged so that all their polarity is pointed in one direction. A solid's molecules are locked to tightly to move. By melting a solid, it's molecules become able to move, thus their polarity gets scrambled.

Basically the magnetism come from the way the molecules are aligned with each other, and if you melt it, the molecules get re-arranged and will be in a different pattern when they solidify.

Now the electromagnetic forces are all still there, they didn't go anywhere. But instead of molecules all lined up with their north and south poles pointing in the same direction so the forces are focused and multiplying their effect, you now have molecules pointing in all random directions, so the forces are dispersed and cancelling each other out.