The last two examples would not be used to describe the corner of a page.

The first two, torn off, and torn away are more or less synonymous. Essentially, they mean the same thing - to be literally torn/ripped from something.

Torn down refers to something being demolished, like a building for example.

e.g. The building was torn down.

Torn apart refers to two or more things being torn from each other. It's often used figuratively to mean being separated from someone you care about.

e.g. When she left for Australia, he felt as if they had been torn apart.

It can also be used literally

e.g. The pages of the book had been torn apart.

The last two examples would not be used to describe the corner of a page.

The first two, torn off, and torn away, are more or less synonymous. Essentially, they mean the same thing - to be literally torn/ripped from something.

Torn down refers to something being demolished, like a structure for example.

e.g. The building was torn down.

Torn apart refers to two or more things being torn from each other. It's often used figuratively to mean being separated from someone you care about.

e.g. When she left for Australia, he felt as if they had been torn apart.

It can also be used literally

e.g. The pages of the book had been torn apart.