Different characters say more than once that a sword forged in a castle is a pretty expensive thing. But nobody, even a totally broke person would loot a defeated enemy to take his weapon and sell it.
Why? Looks like a smart move to me.
Let's rephrase the question: "Why does no one take the murdered victim's sword to sell it?"
CSI doesn't really exist in the Game of Thrones world, so if there are no witnesses to your fight with an enemy, you normally can get away with it. However, carrying the victim's belongings, especially something as valuable as a sword, will implicate you in the killing enough for anyone that wants you hanged to get you hanged as a murderer.
As mentioned in the other answers: If the fight was an honorable duel in public, you are a noble and probably don't need the money from selling a sword, and you certainly don't need a reputation of a looter.
A Lord or an anointed knight would find looting corpses below them. They normally own a fortune so they do not need to sell swords from corpses.
Common soldiers do loot corpses or whatever they fancy in posessions of their victims.
GoT focuses on Knights, Lords, Kings, Theocrats etc. None of them are exactly the sort who would sully their hands in mud to strip weapons off a corpse. The kind of people who would do that are not worth the air-time on screen.
Anyways who is to say that fallen man has better weapons than you? Yours might be way better so there won't be any desire to pick up the junk anyways.
Besides, carrying extra swords and weapons is just extra weight to bear when you're a common soldier. Which is why soldiers prefer to loot more lighter possessions such as valuables, boots, clothes etc. You have to carry your things for hundreds of miles as a soldier. Why burden yourself with something you may or may not get to sell in future?
On a side note, A person is more familiar with his own weapon. Give a man a new sword and he will spend some time trying to familiarize with the thing. For example, if we consider books, Ser Vardis Egen died because Lysa insisted that he should fight with Jon Arryn's sword instead of his own. From AGOT, Catelyn's POV:
The engraved silver blade was beautiful beyond a doubt, but it seemed to Catelyn that Ser Vardis might have been more comfortable with his own sword. Yet she said nothing; she was weary of futile arguments with her sister.
Also it is not normal (Although not unheard of) to wield/carry dual swords as Jaime told Brienne in S03E02:
I never understood why some knights felt the need to carry two swords.
So in conclusion, there are several reasons:
This does happen, albeit in the books more than the TV series.
In Season 6, Episode 7 it is stated that the visitors to the camp are looking for Money, Food or Steel as payment for protection.
Also remember that (most) of the characters that we follow are:
There are many instances in ASOIAF of victors taking weapons from fallen enemies. Few instances of this have been shown on the GoT T.V. show. This is because screen time is valuable and, as has been pointed out in other answers, most victors are lords and knights who do not need to loot corpses. Below are some examples of it happening implicitly or explicitly in the books and show.
Needle is taken from Arya. (Shown in show and books) Mentioned in previous answer.
Ice is taken from Ned Stark. (Implied in show and books)
Dawn is taken by Ned Stark. (Implied in books) Returned by Ned to the Daynes.
Wildlings main source of steel weapons is looting rangers of the nights watch. (Implied in books)
It very much depends on the circumstances. Looting battlefields is well documented throughout the middle ages as was taking prisoners for ransom. It is certainly the case that common soldiers or men at arms might well take the opportunity to strip fallen enemies (or even friends) for nay bits of equipment that they would find immediately useful.
There are also a few other considerations.
Firstly is pure logistics. If you have won a decisive victory and are about to go home you might take anything of any value off a battlefield to sell later but this presents the immediate problem of how you get it back with you. In medieval type warfare logistics were severely limited and a common soldier might struggle to carry just their own arms, equipment and supplies so they need to sort out some way of physically taking any plunder away with them.
In this context soldiers would tend to concentrate on higher value and more portable items like coins, jewellery or even teeth. while you might swap your own sword for a better quality one carrying two or three extras is quickly going to become a major chore.
Also military discipline may make very obvious plunder difficult, whether for moral reasons or just not wanting the baggage train encumbered with loot, especially if the campaign is ongoing.
There is also basic supply and demand. After a major war mediocre swords will be in high supply and low demand and you also need to take them to a market whre they have some value.
All of the above tend to favour local scavengers or more organised and professional outfits who follow battles rather than soldiers who may be a long way from home and more interested in more readily negotiable booty.
Most highborn, merchants and men of wealth had a very distinguished looking swords.
And when it was rare to carry around two swords, carrying two swords would raise suspicions.
There could be a possibility of someone knowing the previous owner of the sword.
There would be rumors about you selling someones sword to the merchant and someone potentially dangerous to you could be fast enough to inquire the merchant about the freshly sold weaponry.
Now there would be more talk about you taking another mans life as long as the loved ones of that fallen man hear nothing of it you are safe, but when they do you will most likely be poisoned or stabbed to death in some alley.
In conclusion: Not taking the sword of a slain man is a life insurance of it's own.
It does happen. Tywin Lannister took Eddard's sword "Ice", melted it and forged "Oathkeeper" and "Widow's Wail". Obviously he did not sell it - valyrian steel is nearly priceless and on top of that he is a Lannister, therefore loaded.
In normal combat that would not happen very often - the swords used by the common soldiers appear fairly standard and thus of little interest to either lords or commoners.
I suspect that the bodies of any lords killed in battle are claimed by the winning lord. I don't have any references at this time, I'll try to find some later tonight.