The button can't be press.

One of the buttons on the remote control isn't working.

The button can't be press.

The button can't be press down.

"Press" and "Press down" are the same? Is it the verb that can omit?

The verb "press" is both transitive and intransitive verb. When used as a transitive verb, you can say "The button can't be pressed (down)". If used as an intransitive verb, you can say "The button can't press (down). However, it's more common to say "The button won't press".