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run-on error?

Remember, it's not true.
Remember. It's not true.

I like the first one, but why is this not a run-on error?

Re: run-on error?

'Remember' is not an independent sentence (no subject).
I would punctuate it as:

Remember - it's not true.

Re: run-on error?

Remember, it's not true.

"Remember" is in fact an independent imperative clause. The subject of an imperative clause is usually omitted, but understood as "you", compare Please sit down; Don't move; Stop!

"Remember" normally requires an object or some other complement, though it has been ellipted in your example.

I don't think it would be unreasonable to drop the comma and treat "it's not true" as a subordinate (dependent) content clause in complement function:

Remember (that) it's not true.

Alternatively, you could treat the two clauses as independent ones and separate them with stronger punctuation such as a colon:

Remember: it's not true.