There are two reasons.
First, "it never exist" is never grammatical: you mean "it never exists".
The second is much more subtle. "Don't tell me it never exists" is grammatical, but is not something an English speaker is likely to say.
I am struggling to explain why this is, but I think the answer is something like this:
"This exists" is not usually a property that can come and go: a thing either exists (or doesn't) for all time, (qualities, for example), or comes into existence at some time, and continues to exist until it no longer does (people, for example).
To negate that, we usually say "doesn't exist". "Never exists" sounds odd, and to me it would imply that we are talking about some odd kind of thing that sometimes exists and sometimes doesn't, and we are saying that this particular thing never does.
An example (somewhat contrived): The King (of a particular country) is something that can exist at some times but not at others. So at present there is a King of Spain, but no King of the United Kingdom.
In that context, I can imagine somebody saying "The King of Spain exists, the King of the UK doesn't exist, and the King of the United States never exists". (It's not a likely thing to say, but it makes sense, sort of).
With a past tense, "never" is slightly different. To say something "never existed" is not unusual - it means that it doesn't exist now, and never has; but it does not have that implication of something that might come and go.
So in summary: "Don't tell me that it never exists" would be grammatical, but unusual, and would imply that "it" was in principle something that could exist and not exist at different times. "Don't tell me that it never existed" would be normal, and not imply that the thing could come and go.