When used as an adverb, あまり has two definitions. In addition to the "not very much" meaning that is restricted to negative sentences, it can also mean "too much" or "excessively", which is the meaning used here. に or にも is often attached when used like this.
The すぎる pattern already conveys the idea of "too much" in the first sentence, so adding あまり simply doubles down on that. That is probably why you were told it adds emphasis. Removing it wouldn't change the meaning of the sentence.
Please don't work too much (hard).
Your second sentence is essentially the same, but the "too much" is conveyed solely by あまり.
Don't push yourself too hard.