Can 'of' be omitted in these sentences?

Solution 1:

Omitting the word "of" in those sentences is a form of slang. It is commonly used, especially in some regions, but it is not proper for formal writing. You can use it if you're writing informally or intentionally using dialect/slang.

I found examples similar to yours in Google Books. The first is from the book Poles 'N' Goals and Hesselink:

Coming out the ground on that balmy night I tried to contemplate what had happened to Celtic, we were top at the end of October, now we had finished so far behind in the league that we didn't even qualify for Europe through it.

And this is from the book At Bully Hills:

Flowing sprays of blood spurted out his hollow neck. The Beast was still alive, his head lay in the sand, black eyes blinking; his mouth gasped for breath. A blood red foul froth foamed at his fangs..

Notice how both sentences come from novels that use a relaxed style. Clarity is the big reason to make sure you're using "of" when describing the relationship between two things. For example, consider the following quote from the Bible.

Take these things out of here! Don’t make my Father’s house a marketplace!

In that sentence, Jesus is saying "dispose of these things, get rid of them." If you change the sentence...

Take these things out here! Don’t make my Father’s house a marketplace!

...he might be understood to be asking that "these things" be brought to where he is, which is a very different meaning.

EDIT: Check out Peter's comment for important context. I was about to adapt the comment into my answer before realizing that I can't improve on how Peter said it.