The playtest Wild Soul barbarian subclass (from Unearthed Arcana: Barbarian and Monk has the Wild Surge feature at 3rd level, causing magic to erupt from the barbarian when they rage.
Row 7 in the Wild Surge table lists the following effect:
Shadows weave around a weapon of your choice you are holding. Until your rage ends, your weapon deals psychic damage instead of its bludgeoning, slashing, or piercing damage, and it gains the light and thrown properties with a normal range of 20 feet and a long range of 60 feet. If you drop the weapon or throw it, the weapon dissipates and reappears in your hand at the end of your turn.
By default, a greatsword (other big sticks are available) has both the two-handed and heavy properties. The description of the two-handed property says:
This weapon requires two hands when you attack with it.
The description of the heavy property says:
Small creatures have disadvantage on attack rolls with heavy Weapons. A heavy weapon’s size and bulk make it too large for a Small creature to use effectively.
But if the barbarian's Wild Surge results in option 7, then it seems it would gain the light property as well:
A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.
Going by the mechanics alone, I see no problem between a weapon being both light and heavy, since a Small creature would have disadvantage and you could dual-wield. But thematically I have a problem with something being light AND heavy since, ya know, they're both opposite ends of the "how heavy" scale. I guess the important question here is: how heavy is it?
The mechanical problems come with light and two-handed. Two-handed sort of rules out dual-wielding unless you have more than two arms (but I've seen this be overruled in official books, such as when a giant wields it in one hand), but light says that it's perfect for dual-wielding. So what happens in this case?
My interpretation is that you can't dual-wield because of the weight and heftyness, but light cancels that out, but I may well be wrong.
The 2-handed property makes the light property irrelevant
As covered in this question about the light property, the only function of the light property is to enable the bonus action attack of two-weapon fighting (TWF), and the light property doesn't necessarily have to do with weight (e.g. some light weapons are heavier than some non-light weapons). However, it's not possible to use TWF with a 2-handed weapon (even if you somehow have more than 2 hands), because the trigger for TWF is:
Hence, you could have 4 arms and be holding a two different 2-handed weapons, but RAW you still couldn't use TWF with them, because you must be holding each weapon in a single hand to use TWF. So, since TWF is unusable with 2-handed weapons, and the only function of the light property is to enable TWF, adding the light property to a 2-handed weapon has no effect at all.
(That being said, the TWF rule is clearly written with the assumption of having exactly 2 hands, and I don't know how much sense it makes to apply it to creatures with different numbers of hands. I don't know of any playable race or class feature that provides more than 2 hands that can wield weapons.)
Heavy, light, and thrown are all mutually compatible
As for a weapon that is both heavy and light, while it seems odd, RAW there is no mechanical conflict between the two properties.
Finally, there is no problem with adding the thrown property to a 2-handed or heavy weapon.