So I've been leaving my dog's toys scattered about; she'll come ask me for fetch or tug by bringing me a toy. I've been recently reading, however, that I shouldn't leave tug ropes out, because she'll chew on them and swallow strings of yarn or whatever it is they're made of and that's bad for her. The thing is, her favorite tug toy is her favorite toy period; she carries it around the house and often asks for tug with it (at least 4-5 times a day). Should I take it away from her and put it up and only get it down for play sessions? She's a shelter dog and I'm trying to give her a sense of security and "home" as well, she was clearly abandoned in a yard before and she hates to be out of my sight.
As long as she doesn't destroy the rope, I wouldn't worry.
Traditional rope toys are made of cotton or a comparable natural fiber because they are intended to be chewed on and ingested in small amounts. As long as you only play tug of war or the dog only carries it around, they are no problem.
The problem arises when the dog actually chews through strands of the toy. If a dog tries to swallow big chunks, she could choke or suffocate on them or they could block her small intestine. And long strands could be a problem because they may not pass the digestive system in one go. If one end is stuck in the stomach and the other end already passed into the small intestine, the intestine can be cinched and cut off from blood flow. This is incredibly painful, and can lead to death because the symptoms are often confused and the problem diagnosed too late. The same problem arises if cats try to eat toys on a long string.
Does she actually obviously damage this rope toy? You say in a comment that she destroyed a different rope toy (that had a ball attached), but if she doesn't show any signs of causing damage to this specific rope toy then she's unlikely to wake up one morning and immediately rip it to a dangerous state. You should monitor it for wear over time and be ready to replace it when it starts wearing out--but that's something you should do for all toys anyway.
If she is obviously damaging it to the point where she can wear through it quickly, then yes you should take it away from her when she's unsupervised...but try to find something else that she likes that's safer. Perhaps if she likes the texture of the rope then she would like some of the textured rubber toys (look for teething toys, those have a lot of texture)... Some of the rubber toys are also tug-type toys.
I actually disagree with Elmy here. If your dog could swallow a large chunk of the rope, they may choke and die.
To prevent this, I would consider:
getting a tougher toy/rope that your dog won't be able to destroy;
keeping your dog under more supervision;
buying alternative toys so you can take the said toy away when you aren't watching your dog and giving it back when you return while still keeping your dog mentally stimulated and happy;
exercising and playing more with your dog so they won't destroy toys out of boredom.