Learn English – “Fall from” vs. “fall off”

This is colloquial grammar used in England, and occurs with other verbs that indicate acquisition; "steal off" and "hear off" are other common examples. It is technically incorrect from a purist's perspective, but is nonetheless in frequent use.

The replacement of "from" with "off" carries a slightly stronger sense of the act of taking, but the construction carries a distinctly lower-class sense and is to be avoided in polite or formal situations.

To take (something) out of the picture is an idiom meaning to remove it from consideration. If you are pondering, for example, whether to trade your sandwich, apple, or cookie to your friend in exchange for his candy bar, you might decide that you would get into too much trouble if you traded your sandwich; so you "take the sandwich out of the picture", reducing your choice to either the apple or the cookies. The sandwich is no longer an option.

To "take (X) off (Y)" generally implies that item X is physically resting on top of item Y. So, to "take this part off the picture" would probably mean that you have a picture, and a part is sitting on the picture, blocking your view of something in the picture. So you move it.

To "take (X) from (Y)" generally implies that X is presently a part of Y and you are going to remove it. So, to "take this part from the picture" means that you have a picture of an something, including a particular item. You intend to remove the item, and re-take the picture without the item present.