I don't think that it would be a bad idea to have dynamically typed databases although it would put a little more strain on developing the actual application instead of having to spend time on making the database.
At Massify, we have 5 different applications with entirely different code bases hitting the same datasources, so now you want me to go and make sure input verification is the same across all of them when I could more easily do it in one source?
It doesn't nullify anything I've said, in fact it proves it!
It would be a HORRIBLE IDEA. Nothing about this is a good idea, if you think it is, please turn in your developer license!
This relationship changes somewhat in smaller organization with a more hand-picked team of course, but even there it is normally less work to set up the schema properly then to make sure every application that ever touches it properly validates every single piece of data for proper type.
There are times when a dynamically typed database makes sense, for instance when it is primarily an object store for an object oriented program. But if you want a relational database with the data integrity and data analysis capabilities it brings to bear, then it probably makes more sense to use a statically typed database.
SQLite is a very well known relational database with dynamic typing, it's quite possible the most deployed relational dB in the world since it runs in everything from browsers to mp3 players and if the data inserts are well controlled, don't really result in any particular problems.
There are a minority of coders who are actually anal enough to get that right 100% of the time in every version of every app that touches the database. But those are not the people who are reckless enough to want to omit types from the schema. If you think doing it the easy and reliable way isn't worth the hassle, you are not actually going to do it the hard way, even if you can convince yourself you will.
And another very important point, applications errors can be hard to fix, but Terabytes of inconsistent data over the years that your business depends on, now that's nightmare.