[RPG] Does PC weight have a mechanical effect

Carrying another PC, speed reduction

Depending on which version of the encumbrance rules your table is using, if you end up in a situation where one PC has to carry another PC (for example, when fleeing a horde of creatures when you are low on resources) then the weight will allow the DM and the Player to determine the level of encumbrance, and thus movement penalty, that carrying the {unconscious, paralyzed, sleeping, whatever} PC levies on the carrying PC.

Lifting and Carrying (Page 63, Basic Rules).

Carrying Capacity.
Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don’t usually have to worry about it. Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying capacity, your speed drops to 5 feet.

Size and Strength. Larger creatures can bear more weight, whereas Tiny creatures can carry less. For each size category above Medium, double the creature’s carrying capacity and the amount it can push, drag, or lift. For a Tiny creature, halve these weights.

A Goliath carrying a paralyzed gnome? Not a problem.
A halfling trying to carry a dwarf or another halfling? The added weight could slow them down considerably with a pack of wolves on their trail ...

Variant: Encumbrance (Basic Rules, p. 63)

If you carry weight in excess of 5 times your Strength score, you are encumbered, which means your speed drops by 10 feet.

If you carry weight in excess of 10 times your Strength score, up to your maximum carrying capacity, you are instead heavily encumbered, which means your speed drops by 20 feet and you have disadvantage on ability checks, attack rolls, and saving throws that use Strength, Dexterity, or Constitution.

When using this rule, the disadvantage to things like ability checks (to climb up and down mountains while carrying a party member) could create a significant problem for the player to solve.

Can this mule carry us?

The carrying capacity of mounts is linked to weight; if the party is trying to get multiple wounded or incapacitated characters carried by a mount, their weight may create limitations. (PHB, p. 53-54) Similar situational considerations as mentioned above.

Can I pick up that gnome with my mage hand spell?

Depends on what the gnome weighs. Mage Hand is limited to 10 pounds. (thanks @mattdm)

How many characters can the wizard carry on their Tenser's Floating Disc?

TFD exploits and uses have a long history in D&D, but it has a weight limit (500 pounds).

Weight limits also apply to magic items like the Broom of Flying (slower flying speed if carrying more than 200 pounds), Rope of Climbing (3000 pounds), and Immovable Rod (8000 pounds).

Does this need homebrew?

I don't think so, but that's really a matter for your table to decide upon. A DM ruling with the above in mind should be able to handle most situations. (On the other hand, homebrew away if your table finds that fun).

Carrying loot and treasure, or the disabled character

If you find a bag of silver pieces ... or copper pieces ... or gold pieces ...

A standard coin weighs about a third of an ounce, so fifty coins weigh a pound

... but you are also being pursued by bandits, wolves, whomever you stole the coins from, etc ... how much the party can carry before being slowed down can become a decision point: how much treasure does one abandon to enable escape, or, with this much treasure after that long dungeon crawl, does the party fight (and if need be fight to the death) since it's the major haul?

This kind of decision has been part of the game since the 10 GP = 1 pound scheme in Original D&D three books published in 1974. It's can still arise as an issue, if the situation is that the party has a lot of heavy loot but needs to move quickly for an in-game reason. How hard your DM leans on this impact of weight, and the trade off between treasure and someone who needs to be carried, can result in some unlooked for role play and party interaction.

In some parties, one of the PCs may be unceremoniously dumped by the others who care for the loot more than their party member!
Yes, it can happen. But it doesn't have to. The strength of the bonds between party members informs this.

Similarly, if the "carried" character is an NPC, and the party drops the NPC to reduce weight, that too can lead to some interesting consequences and role playing.