Because tearing was required for the repair of holes in the curtains of the Tabernacle it falls under prohibited acts.

The prohibition of tearing is connected to the prohibition of sewing. The definition of tearing also includes cutting ay soft material and separating by peeling or cutting apart two items that had been sewn or glued together.

The prohibition comes into force most commonly with the opening of packages and wrappers.

The Biblical definition of tearing only restricted tearing for the purpose of creating, specifically for sewing. In Rabbinic times, the prohibition was expanded to include destructive tearing as well.  The rabbis feared that what was torn during Shabbat might be mended following Shabbat therefore violating the restrictions on sewing.

The limitation does not prevent the tearing of an item to enhance the enjoyment of Shabbat. It is therefore permissible to open a food wrapper in a destructive manner.  The package must be opened in such a way that it can not be reused or resealed for food storage following the meal.

Bottle caps may be removed provide they are not the type that break during removal.  Ring tab tops are not permitted to be opened unless punctured first.  

A sealed carton may not be opened unless it is done in a destructive manner and it rendered unfit for storage following.  When opening a cardboard box, the box likewise must be destroyed in the process without tearing through the writing or images on the box.

Pushing a straw through the hole of a juice box is permitted as it destroys the packaging from further use.

Tearing toilet paper is forbidden as is tearing or separating tissues from one another.  Likewise one can not open a sealed letter even in a destructive fashion.  Napkins, paper towels and toilet paper may be used even though it is possible for them to tear during normal usage. It is however not permissible to tear them from the roll.

Pages of a book may not be separated on Shabbat if the attachment occurred in the book binding process.  When food causes the pages to stick together inadvertently, the pages may be separated provided they do not rip.

Adapted from The Shabbas Home by Rabbi Simcha Bunim Cohen