There are not. The game assumes that all characters are taking care of their equipment (and studying and practicing and so on) off-screen, and basically assumes that there are never any particular troubles with this. The rules work this way because the authors assume that the characters have much bigger, more interesting troubles to be concerned about.
The only way we’re likely to see such rules from WotC is, maybe, as part of some limited optional variant rule. Such variant rules are rarely comprehensive or complete; rather they just offer a few ideas for how you might change the game some, and then expects DMs to figure out the details.
Which is what your DM here is doing. He is attempting to incorporate a (realistic) detail that the rules themselves ignore. The rules themselves encourage that kind of customization, but it is then the responsibility of the DM to fully flesh out those rules and clearly explain them to the players (at least insofar as the player characters would know them).
I would further add that it is his responsibility to fully understand the situation: why the rules are the way they are, how this change is going to affect things, how other things should change for consistency, and what the effects those will have, and so on. Homing in on one particular detail while ignoring the rest is a mistake that leads to an inconsistent game. And changing everything to pay greater attention to maintenance is a huge change with significant effects, and by definition is also taking away attention from elsewhere. The DM should be aware of all of this when making changes.
So you will have to ask the DM for the rules he’s making up. I would suggest, at the very least, that a magic bow should be exempt from such concerns; magic items are generally far, far tougher than their non-magic counterparts, and such an exemption would mean that dedicated archers should be able to cease to worry about this problem at relatively low levels.
Why isn’t this part of the game?
What follows is a somewhat-lengthy, somewhat-tangential discussion of why such a rule is not part of the game, and why I say it almost-certainly never will be (outside of optional variant rules). This may be of interest to you, and it may also be something you would be interested in showing to your DM (such is my hope, at any rate). As I’ve already stated, the DM should be taking on a lot of responsibility for understanding the game and understanding the significance of his changes when he makes them. This may help him understand that.
Despite some claims, D&D is not a be-all, end-all system perfect for every sort of game, or even every sort of fantasy game, or even every sort of medieval fantasy game. Its rules, in aggregate, paint the picture of a very certain sort of fantasy game, which come from its roots and is hinted at by the name. In essence, D&D is first-and-foremost designed to be for dungeon-delving and dragon-slaying.
Now, to be clear, this priority may stem from D&D’s history, but that does not mean the game hasn’t changed over the years or across the editions. What it means to go out dungeon-delving or dragon-slaying has changed some—and Wizards of the Coast has tended more-or-less to try to stick to modern expectations for those activities rather than necessarily trying to hew as close as possible to the original vision. 5e is something of a reboot for them, after 3e (perhaps unintentionally) strayed from those ideals and 4e embraced that straying to a degree that alienated significant portions of the audience, but nonetheless it has plenty of modern sensibilities to it.
The reason I bring this up is because this kind of maintenance concern is precisely the kind of thing that (modern, WotC) D&D has absolutely and categorically ignored. It is not a part of the fifth edition, and that choice seemed very much intentional on Wizards’ part. It’s impossible to say if that reflects some market data they have or simply the personal preferences of their team, but the fact remains that 5e has set its abstraction threshold firmly above the point where one would be concerned about an always-strung bow.
“Abstraction threshold” is an important term in game design, and refers to the degree to which the game attempts to emulate reality. All games, by definition, must simplify reality to be playable, and simplification is often done through abstraction—conflating things that are really distinct, ignoring caveats, qualifications, or limitations, smoothing out edge cases. This is important to making a game; where you choose your abstraction threshold defines a whole lot about what kind of game you have and what sort of stories you can tell playing it.
And, to quote Heinlein, there ain’t no such thing as a free lunch: changing the abstraction threshold focuses more attention on one thing, at the cost of other things. Attention and abstraction is not strictly zero-sum, but everyone’s time and attention is limited. Every rule demands time and attention, so the more demands you have in one area, the less will be possible to devote to another.
WotC D&D’s abstraction threshold is set to encourage “epic” narratives. Caring for bowstrings is not something that is part of that narrative. Legendary bow shots are what that narrative wants to relate. Having rules about bowstring maintenance draws attention away from epic bowshots—time and attention must be paid to maintaining the bowstring, and that will necessarily come from time that could have been spent setting up, attempting, and making legendary bow shots. And so I assert that bowstring care will never find its way into the rules of the game. Other games have different abstraction thresholds, and even earlier editions of D&D may have had much greater focus on this kind of maintenance and preparation, and may very well have rules for damage to a bowstring if left strung.