Formal fallacy that describes that lack of observation does not prove non-existence? Is there a formal fallacy to describe lack of an observation is not proof that it does not exist, or lack of an occurrence is not proof that it can never happen?
 A: 
Is there a formal fallacy to describe that lack of an observation is not proof that it does not exist, or that lack of an occurrence is not proof that it can never happen?

Absence of evidence is not evidence of absence. -Carl Sagan(?)
You are looking for an insufficient-evidence fallacy; here are two, in the context of your requirement:

*

*appeal to ignorance: something is false because it hasn't been proven true;

*hasty generalisation: a biased/small sample in which something is false is enough to conclude that it is generally false.

However, because the above are weak inductive arguments rather than deductive reasoning, they are not formal fallacies, which describe defective logical structures.
A: This is the Appeal to Ignorance fallacy: absence of evidence is not evidence of absence.   It is not considered a formal fallacy though.
A: In most general terms , we can state it Mathematically / logically / formally like this :
Let X be the Universe (eg Integers or real numbers or .... )
Let Y be the non-empty subset which DOES NOT have Property P.
$ Y = \{ x \in X : \lnot P(x) \} $
Now , $ \forall y \in Y , \lnot P(y) \implies \forall x \in X , \lnot P(x) $ is the fallacy.
We are not able to observer Property P in Y , hence we think (over generalizing) that it must be true every where , in X.
We must not take Examples from Y to over generalize to X.
Let Z be some arbitrary non-empty subset of Y.
Here too , $ \forall z \in Z , \lnot P(z) \implies \forall x \in X , \lnot P(x) $ is the fallacy.
We are not able to observer Property P in some small Collection Z , hence we think (over generalizing)  that it must be true every where , in X.
We must not take Examples from Z to over generalize to X.
Example 1 (lack of observation) :
Let a Continent have many countries , where the Presidents are all over the age of 70.
We may think , that in all continents , in all countries , the Presidents are all over the age of 70.
We then check in the nearest Continent , & observe the same , hence , with these Examples , we conclude on this "fact".
When we then check more Continents , we see that there are Countries with young Presidents , hence the fallacy gets exposed.
Example 2 (wrong observation) :
We have checked Polynomial Equations like $Ax+B=0$ & Determined the Solution.
We have checked Polynomial Equations like $Ax^2+Bx+C=0$ & Determined the Solution.
We may think , we can always get the Solution in "Closed-form" radicals.
We then check Polynomial Equations like $Ax^3+Bx^2+Cx+D=0$ & Determined the Solution.
Hence , with these Examples , we "conclude" that all Polynomials Equations ( including Degree 4 ) will have the Solution in "Closed-form" radicals.
Then Galois (along with Abel & Ruffini) shows that it is Impossible , there by exposing the fallacy.
In Summary , we can not observe some small set to then take these Examples to  over generalize to the universal Set.
More here :
Proof By Example
