I'm not a Catholic, but I have attended Catholic Mass and I've been acquainted with the practice of regular mass- that a certain week of the year has the same readings each year.
Is the entirety of the Bible read throughout the year during mass? If not, what passages (or books) are never included?
The Revised Common Lectionary (by far the most common, used by Roman Catholics, Episcopalians, Methodists, and others) is a three year cycle, and if you attend every Sunday, you are guaranteed to hear selections from every book of the canon, but not necessarily the entirety of each book.
That said, there are several texts you will hear repeated, either for their suitability to certain parts of the church year, or because of their relative importance.
Finally, if you read the daily lectionary (a 2-year cycle), you will cover much more of the bible, but there will still be some parts (1 Chronicles 1-10, I'm looking at you!) that will still be missed.
As a Baptist turned Episcopalian turned back Baptist, I can say that I was surprised at how much more Scripture one hears in a lectionary setting. It is a nice discipline, but it is not exactly the same as reading the Bible through in a year.
There are two lectionaries at play. The Revised Common Lectionary, or rather a slightly modified version of it, which is a 3-year cycle, and 2-year weekday lectionary. Of course, the lectionary does change sometimes. The Catholic lectionary being used today is not the same as the one in use prior to the 1970s. There are also some slight differences between American and Canadian lectionaries, etc.
You might be interested in this site that compares various different Catholic lectionaries (it give the verse references) and gives statistics on quantity of verses included or percentage of the how much of each book of the Bible is included in each Catholic lectionary reviewed: The Catholic Lectionary Website compiled by Felix Just, S.J., Ph.D.
For instance, the statistics from that site show that the pre-Vatican II lectionary used 16.5% of the New Testament while the current Sunday lectionary uses 40.8% and the current weekday lectionary uses 71.5%. The pre-Vatican II lectionary used 1.0% of the Old Testament while the current Sunday lectionary uses 3.7% and the current weekday lectionary uses 13.5%.
No. Some texts are never read.
A complete list of all Biblical texts read in any type of mass is here (in German). You can click on a book of the Bible and you get a list of the verses ever used in any of the masses of the three cycles of the liturgical calendar.
(source: this answer)
