The news around this, this thread, and similar on Reddit seem to be assuming a jump that isn't supported in this paper.
"The residue remaining after the rinse cycle varies between 1:250 to 1:667 for the dishwasher detergent and 1:2,000 to 1:10,000 dilutions for the rinse aid."
This is actually an uncited claim in the paper, and the paper authors did not test it.
"In the present study, we hypothesized that if the detergent and rinse aid residues are not completely removed, once dry, they may remain on the surface of the dishware."
But then they go on to only test what would happen to gut lining cells if most dishes most of the time had residue on them in such concentrations as they assume would be present, transfer to the food, and make it past the stomach.
This line of research really should have started with something like,
"we swabbed 50 recently washed and dried dishes from 25 different restaurants using different dish-washing machines. Here is the average concentration of detergent and rinse aid residue on the dry dishes".
I get that's not the study they wanted to do, of course; who wants to spend hours and days swabbing dry dishes when you have a decent reason to play with new biotech?
But that means the only thing this paper actually looked at is: IF there is residue on the dry dishes, and IF that remains in the assumed concentrations by the time you eat off of it, and IF all of that residue ends up attached to your food, and IF all those chemicals make it past your stomach, let's see what that might do to your gut lining.
Not a bad idea for a study, but definitely not what people on the internet are taking this to be.
The news articles are promoting that unsupported claim to an assumed fact that there is chemical residue on all the dishes people are eating from and that this is destroying people's gut lining. Most of the headlines around this are something to the effect of: "Commercial Dishwashers Destroy Protective Layer in Gut".
Which is pure sensationalism, the paper authors never claim this and this idea is not supported by present evidence.