Problems with cleanliness
One of the biggest problems with 'perfect' cleanliness and sterilization of all surfaces would be effect on gut flora. Here is an excellent summary of how gut flora is formed. Let me excerpt a few bits:
Environmental sources of the bacteria that are first to arrive into
the gut of newborns are believed to be the vagina, skin, and feces of
the mother, providing an inoculum that is a mix of intestinal and
nonintestinal adapted species.
Quite recently some changes in the composition of these first
colonizers have been reported, and these have been linked to more
stringent hygienic conditions of delivery.
After the first 2 wk of life, it seems that a quite stable,
feeding-related (breast vs. formula milk) microflora is established
and stably maintained.
And in conclusion:
All the above suggest that our knowledge on postnatal microbial
development is far from complete and that the next research efforts
should be geared to illuminate our knowledge of this field, including
provision of a timeline for the presence of strictly anaerobic
I add these excerpts to show that there is a lot going on microbially in the human body, and we are only scratching the surface of what we know about it. Basing this on a sealed city with extreme germ-removal capability, this suggests a few different ways to enact a plot in your story:
We made some gut flora extinct and now we can't eat things
In our zeal for cleanliness, we made some gut flora extinct by sterilizing women as they give birth, or even better, growing the baby in tubes and purposefully transferring bacterial culture to the newborn. Something went wrong and now we have caused the extinction of a major type of gut flora and now we can't metabolize something important, like the amino acid lysine, or Vitamin E, or something. Horrible starvation, scurvy or rickets ensue.
We made gut flora extinct and now everyone has diabetes
A well known autoimmune disease is type 1 diabetes. This is caused by the autoimmune destruction of beta cells in the pancreas that produce insulin. The cause is unknown, but it is believed to have a combination of genetic and environmental factors. Well extinction of some gut flora could be an environmental factor, so lets say that all new children born after this gut flora extinction event get type 1 diabetes within a few years.
We tried to culture gut flora and now its mutated and killing us
Type 2 diabetes is generally caused by being overweight and getting too much sugar or saturated or trans fats. If the gut flora mutates and is now processing food INTO trans-fats, then we are going to have an immense and unhealthy amount of it in our diet. Most people get type 2 diabetes by the time they are twenty.
We tried to culture gut flora, and now everyone has psoriasis
Going with the theme of trying to keep gut flora in culture, psoriasis (another auto-immune disease) has been known to be caused by gut colonization by Staphylococcus, Malassezia, or Candida. If we get something into our gut flora culture that we can't get out (can you imagine trying to remove all the Candida from 100 trillion bacteria in a person's gut!!), then we have no choice but to transfer it to everyone, knowing that at least psoriasis is better than starvation.