I think "error" can be an anagram indicator, but there are some caveats.
Run Time Error (4)
I would be fine with this. "Error" is applicable to the previous word in the same way that "news organization" might give SEWN; it's an error of the word "time", so a "time error". (The strictest of Ximenians typically don't allow nouns as anagram indicators at all. I'm fairly strict, but even I'm fine with this.)
Error: Please get a rest (6)
This one doesn't work as well. I can't think of any analogous situations that I would accept -- I don't think "error: X" means "anagram of X". (Also, "get a rest" is not a synonym for ASLEEP; "getting rest" is.)
Error 106E: Something Wrong (4)
And your third has the same major problem as the second. Anagram indicators that are nouns typically can't go before the text they change; there's not really a feasible way for 'error' to modify the word coming after it with the correct meaning.
For example, "error message" means "message about an error [in some other program]", not "message that itself has an error". I don't think there's a way to finish the phrase "error ___" to mean "___ that has(/is) an error" - and that would be the required reading to get "error" to work as an anagram indicator.
In addition, your third clue is an indirect anagram, which many publishers and setters do not accept. A clue like
Grotesque monster is tough (5)
giving the answer HARDY (anagram of "hydra") is generally not fair, because there are way too many possible monsters. This type of construction -- where you take a synonym (or do some other wordplay) and then anagram -- is called an "indirect anagram". It's generally considered a Very Bad Thing™, because it's unfair to the solver. It's not as bad in this case as it could be, but "avoid indirect anagrams" is a good general rule.