You kind of have two questions going here that need to be handled separately, but in general there is no rule in place to increase the damage dealt by an improvised or manufactured weapon based purely on extreme weight.
Oversized and super-heavy manufactured weapons
Arguably, a manufactured weapon that strays far from the specified stats given in the book, such as your theoretical 20-lb warhammer, would become an improvised weapon, or at best be treated as a different weapon entirely (such as a maul). It could also be treated as a weapon sized for a larger creature; since doubling the size of a creature or object multiplies its weight by 8, we could declare that a 20-lb warhammer is sized for a Large creature (as 20 is pretty close to 16).
The Dungeon Master's Guide has a section on p. 278 that talks about big monsters using oversized weapons, and makes a side-reference that could apply to characters trying to use such weapons (one of those nearly-hidden rules we love so much!) --
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it's Huge, and quadruple the weapon dice if it's Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3d12 slashing damage (plus its Strength bonus), instead of the normal 1d12.
A creature has disadvantage on attack rolls with a weapon that is sized for a larger attacker. You can rule that a weapon sized for an attacker two or more sizes larger is too big for the creature to use at all.
So if you really want to use an oversized weapon, you could double the damage dice and roll with disadvantage. I wouldn't recommend it, but you could, in theory, do so, and it certainly has the backing of rules text.
Ironically, the one place where a player character is likely to come into contact with oversized weapons -- the enlarge spell -- doesn't follow this rule at all; while it increases your size (and your weapon's) by one category, it instead adds +1d4 damage to the weapon, regardless of its original damage dice. I would assume they found that doubling the damage dice was too strong for a 2nd level spell, but this does add a bit of a confounding factor to questions about how to handle oversized weapons.
A few side notes:
There is no similar rule listed for reducing weapon sizes for characters smaller than Small, but judging by the Sprite, it seems likely that Tiny manufactured weapons just deal 1 damage regardless of type.
There is no reference to characters with extreme strength being able to wield oversized or heavy weapons better than weaker characters, though -- for example, having Strength 20 does not empower a character to use a greatsword one-handed, in case "use a greatsword" seems like an alternative to "use a really big longsword".
A DM could definitely just rule that a creature could treat an outsized weapon as a correctly-sized different weapon. Wielding a giant's shortsword as a human-sized greatsword is probably just fine, and fits in with the classic examples like Bilbo Baggins using Sting as a hobbit-sized short-sword even though it was a mere dagger for an elf. It's hard to argue with Tolkien. But on the other hand, D&D treats Small vs Medium somewhat differently than Small vs Tiny or Medium vs Large, so that may not be a very good basis for comparison.
Oversized improvised weapons
Improvised weapons depend heavily on the DM's rulings at the table, so your results may vary depending on who's running the game.
Officially, all improvised weapons deal 1d4 damage, and that's that. A DM is likely to declare that some objects are simply too large and heavy to even count as a weapon, no matter how strong your character is. For example, if somebody tried to use a 20-person table as a weapon, I'd just have to turn that down. You can use it in a fight but it's not going to count as a weapon, swinging around like a huge club; I'd probably rule that you could use the table to Shove an enemy and get advantage on the contested Strength check.
Using absurdly sized objects as weapons is probably a good time for a little off-the-cuff ruling. Going by the 'weapons for large-sized creatures' rule above, I might decide that using a bag full of rocks deals 2d4 damage, but imposes disadvantage on the attack roll. Or I could ignore the weapon-size rule and instead say the bag counts as an improvised maul, dealing 2d6 damage, but it's an awkward and clumsy as a weapon and thus imposes disadvantage -- that's just a ruling made in the moment, off the top of my head, but it feels pretty fair to me, and that's the basis a lot of DMs are going to end up using if you suddenly decide a bag of rocks is your best choice in the moment.