It varies. We've never concerned ourselves too much with encumbrance, but they still have an awareness of copper in particular being bulkier than it's worth to lug back to town. When the PC's are lower level every copper is ultimately worth it. But as they gain levels and treasures get larger the copper and silver rapidly lose meanigful value. By the time they are getting treasures with tens of thousands of gold coin they are unlikely to care about 10,000 cp, yet they WILL still take them unless I mention that they are getting into encumbrance issues. Then the copper is the first to be jettisoned without batting an eyelash.
Even when they do take it, the copper gets spent first. Meals, common supplies, inn rooms, etc. are all paid in copper. If nothing else the copper pays taxes and the fees to EXCHANGE it for larger coins or gems.
Another important factor is whether or not there is a party fund. Most times the PC's as a group eventually come to keep a seperate account from their own personal funds. This party fund is used to buy things that the party as a whole requires - passage on a ship, emergency resurrections, buying information, and even throwing parties. It also tends to include things obtained by the party as a group that retain value for what they ARE such as rope, tents, rations, and caches of weapons and armor (even magical equipment). Rather than trying to sell them off the party will store them somewhere and then have at least the capability (whether used or not) to outfit a fighting force on quick notice or to replace their own destroyed or stolen equipment until better equipment can be obtained.
With a party fund nobody really carries the "useless" copper or silver - THAT'S the coinage that is the first to go into the party fund. Given the mundane things that the party fund is generally used for it actually works better that it is filled with mostly low-denomination coins.
And yet some parties, particularly in older editions where the copper was even less valuable, would simply leave the copper lie (and then one enterprising PC funded a highly successful criminal organization in part by retrieving the copper the other PC's left behind).
The final factor is whether the PC's eventually obtain magical means of carrying loot like portable holes or bags of holding. As soon as they do the question is moot - EVERYTHING NOT NAILED DOWN gets shoveled into the magical containers and sorted out later all-but-regardless of bulk or value.