They wont swallow anything harmful. I would use as a bottom to their cage an old bathtowel as you can wash/reuse it all the time instead of buying bedding. In side their house you can stick old cotton T-Shirts and similar items. They will snuggle into these items and again you can wash them out and reuse them when the urine smell gets into them.
Tissues are perfect. They love nesting out of it and they won't eat it. As long as it's not scented or anything, it wouldn't even be harmful if they did inject some. Hay would actually be more harmful than tissues as rats aren't build to digest it and hay really isn't all that comfy.
Hay can be really dusty too plus it has the potential to mold if it gets wet. If you want a litter type bedding, go with paper or aspen. A lot of people here use fleece for cage liners rather than wood or paper bedding (me being one of them). You can check out the cage pic thread to get some ideas.
If you get good quality Hay is a lovely nest for them though it's not hugely absorbant so you need to change it now and then
saying that i find tissue gets smelly in a few days to. Rats also rarely choose to do more than nibble a few peices and it's great for enrichment s they can dig in it quite well too. Just look for low dust and green in possible.
tissue is also fine just don't ever use cotton wool bedding sold for hamsters. It's not good stuff and rats can eat boys and get it stuck in thete digestive system.
In the floor I'd recommend a loose substrate like aspen in the us or shredded card and or hemp bedding in the UK. lose substrates encourage natural behaviour like foraging abs digging. cage liners look pretty bit aren't as absorbant and aren't as fun for the rats (is more for the owners than the rats).
I was under the impression the OP was asking about extra bedding for for the rats to nest with rather than the main bedding. I apologize for the misunderstanding.
Yes, tissue has no odor control and will smell quickly so it's not good to use as bedding by itself. I mix it with Kaytee Clean and Cozy which is a wonderful paper based bedding. Very absorbent and keeps the smell down.