Do I take the teddy bear away or continue to let her care for it?

If she says it has died, take it away (or whatever she seems to want).
You can bring it back before long (maybe the next day) because she probably won't remember. Nothing wrong with her keeping the Teddy if it seems to bring her comfort.

Probably let her keep teddy since it still brings her joy. On bad days, it might be best to divert her attention to other topics and activities.

Mom’s been sleeping w a big stuffed beagle since my dad passed. When she moved to AL i replaced it with a clean one. She also really enjoys her beanie babies. She has a sofa arm tray with pockets for glasses, nail files, etc. and the tray on top holds her pens and snacks… then she discovered that the beanies sit nicely
up there and are close so she can hold them and talk to them. They make her smile and laugh. She often says ‘ Have you seen this? His big eyes are so funny.’ ( i pretend they are new to me even tho i brought them awhle back) So very thankful her level of dementia has made her childlike and happy alot with simple things like toys, ice cream, hersey kisses and bananas.

Don't take it away. Have you tried playing along and "rescuing" the baby by giving it medicine or telling her it was just sleeping? See how she reacts to that, but do it in a very calm and reassuring way.

My 99-yr old aunt with mod/sev dementia helped raise me from birth but never had children of her own. She has fixated on a llama-like multi-colored stuffed animal as her baby. The irony is that she and her housemate sister made and purchased numerous dolls over the years, many that even look just like newborns. Her house is full of them in every room. But those she is not interested in. Just the weird rainbow colored llama. Perhaps because it is very soft and not hard and plastic-y like the dolls.

Let her keep the teddy bear if it brings her comfort and joy. She's obviously out if it if she believes it's her child, so taking it away would be cruel.
All the responses here are right about when she thinks it has died. Just reassure her that it isn't so.

It's just mind boggling how quickly things change with dementia. One moment the teddy bear is an object of comfort, the next moment it's an object of fear & sadness. There's no good answer here, as there is no good answer with the angst & torment of dementia in general, except to say leave the teddy bear with your Aunt. Reassure her, as the others have said, that the teddy is alive and well when she fears he's stopped breathing, and distract her onto another topic. Until the next time.

There is always 'something for you to do' for your Aunt when dementia is involved; there is always comfort to offer or a shoulder for her to cry on. Of course nothing is going to stop or slow down her decline, until the blessed peace of her transition occurs. But in the meantime, offer her all the comfort you can, and have been. A baby doll is often a great source of comfort for dementia patients, so it may be worth your while to buy one for your Aunt to see how she likes it.

Best of luck.

No, don't take this object of comfort away.
Grandma1954 has another good idea, but she may still prefer her teddy bear. If she does have times when she cries over the bear and what she imagines it is only to provide her comfort, to tell her you are sorry, ask her if there is anything you can do, comfort her.
Do know that for her this is very real. Taking it away would likely be very painful for her. It was the opinion of Oliver Sack who lifelong studied the mind, that their worlds are very real, just so different and alien from our own that we can't understand that. So our deficits are perhaps as severe as hers? Many things are beyond explanation and understanding. This is but one of them.

Leave the teddy with her. It's keeping her occupied & has something to cuddle

There are "life like" weighted dolls that might be to her liking.
There are animals that have features like heart beats and they make noises, cats purr.
But it is possible that none of these would work or give her comfort like the teddy bear she has. (try telling a parent that lost a child that another can replace the one that died)
If you are with her when the "teddy bear dies" try to reassure her that it is just sleeping and the breathing and heart beat slows when they are sleeping. Tell her that the Teddy Bear will be ok when he/she wakes up. Then try to redirect, "let's go get a snack ready for Teddy when he wakes up and we can have some tea now."

There is nothing for you to do. Nothing will stop or slow down her decline.

My husband likes two stuffed toys here-which I introduced to him a few months ago, thinking that would help him. It does, he has them talking in different accents, hanging out, teasing the cats and each other. He hasn't paid much attention to them recently-but I make sure to keep them close to his bed and might ask him soon how are they doing. I've been thinking about adding other stuffed animals to the mix, but it might be too much for my husband to intergrate into his two toys. In fact, one of the web sites that sells clothing for dementia patients also has over sized teddy bears and I wonder if these are comfort toys for dementia patients.
Also, if possible when your aunt's teddy bear has recovered from a near death experience, is it possible to gift the bear with a scarf or some sort of gift? Reinforcing a positive outcome for your aunt might reduce the possibility of a reccuring teddy bear terminal illness and lessen her stress overall.

There are teddy bears with heartbeat recorders.

I wouldn't take her bear away, it gives her too much pleasure. Is there a heartbeat bear that is close to what she has?

Never, never take the bear away. Reassure her that he's fine, but never take that connection away from her. There will be fewer and fewer connections for her as time goes on, and the bear may be the last one she keeps.

I wish my hubby had a teddy bear. He is currently fixated on a long-handled shoehorn. He carries it with him and frets if he sets it down and forgets where it is. He actually sleeps with it clutched in his hand.

I’m glad she finds joy with the bear. That’s a gift for you both. If she becomes distraught say something like “no silly, she’s napping, you know babies sleep deeply” and distract her. Hopefully she’ll be satisfied and move on

I would reassure her that the baby is just sleeping and she has to calm down or she will upset the baby.

My grannie loved her baby doll and we always put her down for her nap so we could cruise around the NH and visit. We bought new clothes, changed her diaper when it got saggy from to much loving and made a fuss about how big she was getting, just like a real baby. It worked and grannie was satisfied that all was well when we went cruising.

No you don't take her teddy bear away, as she finds great comfort and joy taking care of it. Many people with dementia(women in particular)love to carry around a baby doll or in your aunts case a teddy bear, and love on them, and care for them. You will see that a lot if you were to ever visit a memory care facility.
I think it's quite sweet to see, and it does no one any harm.
Perhaps next time she says that it has died, you can just go along with things and tell her that you will try giving it mouth to mouth, or that you will take it to the doctor.
Bless you for looking after your aunt.