Some ranged weapons have a pretty short maximum range, like nets, for example (15 feet). Is there a way to increase a ranged weapon's maximum range? Any increase, as little as 5ft, interests me.
There is currently no way to increase the maximum range of a weapon. Which is to say there is no such thing within the Player's Handbook, Monster Manual, Dungeon Master's Guide, Sword Coast Adventurer's Guide, Volo's Guide to Monsters, Xanathar's Guide to Everything, or any of the currently published Unearthed Arcana articles.
The closest thing to what you're looking for is this ability from the Sharpshooter feat:
Attacking at long range doesn't impose disadvantage on your ranged weapon attack rolls.
Which would allow you to attack up to 15 feet without disadvantage.
If you're specifically looking to increase the range of nets, there is this somewhat dodgy trick:
An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.
So if you can find an object that doesn't resemble a net, but your DM accepts that it acts like a net (maybe a lasso or a bola?), you might be able to have a 60 foot range, 1d4 damage weapon that does the same thing as a net. If you're going to try this, I strongly recommend taking the Tavern Brawler feat (which includes proficiency with improvised weapons) to make your DM feel a little bit better about it.
Alternatively, you could ask your DM to work with you and homebrew something - a homebrew feat that increases maximum range, a homebrew lasso, or something else that lets you achieve your goals.
I added a comment about the kind of distance you're trying to tack on to the stated range, but I'll give a couple examples as an answer also.
Here is what the SRD says about ranged attacks:
You can make ranged attacks only against targets within a specified range.
If a ranged attack, such as one made with a spell, has a single range, you can’t attack a target beyond this range.
Some ranged attacks, such as those made with a longbow or a shortbow, have two ranges. The smaller number is the normal range, and the larger number is the long range. Your attack roll has disadvantage when your target is beyond normal range, and you can’t attack a target beyond the long range
So I going to assume you want to go even beyond the longer range (Spells are more fixed so I am not going to cover them).
So bow weapons already take into account a longer range. You could also apply this to all other ranged weapons and give them an extra distance at a disadvantage. How far it flies and is useful is up to the DM.
These things are only going farther because of extra effort on the characters part; putting their whole body into it as opposed to a normal throw. You could also put an extra DC10 on the effort. If both to-hit throws are less than 10 (and miss their target) then the character is off balance and all attacks against them for the rest of the round are at an advantage.
Steve the barbarian heaves the net with all of his might, straining all the muscles in his legs and arms to ensnare the fleeing goblin.
'Rolls a 5 and 7'
Despite his best effort, the net goes wide and the goblin gets away. Steve is unsteady on his feet from contorting his body so much.
Another DM-choice would be allowing modified items to go farther but for less effect. A smaller net can go farther (takes less strength) but only entangles for one turn maximum. The smaller axe can be flung farther, but only does 1d2 damage. Both would be enough to break concentration or otherwise effect the target, but not nearly as much as a full sized option.