Step 1: Clarify what needs to be checked.
We are given a $3 \times 3$ matrix $A$ and are asked to examine four statements involving:
the row-reduced echelon form (RREF) of $A$,
the eigenvalues of $A$, and
the solvability of the system $Ax=b$.
Our goal is to determine which statement is incorrect.
Step 2: Examine statement (A).
If the RREF of $A$ is the identity matrix $I_3$, then $A$ is invertible and has full rank.
An invertible matrix cannot have zero as an eigenvalue, because a zero eigenvalue would imply singularity.
Therefore, this statement is correct.
Step 3: Examine statement (B).
This statement claims that if zero is not an eigenvalue of $A$, then the RREF of $A$ must be $I_3$.
While the absence of a zero eigenvalue does imply that $A$ is invertible, it does not mean that the matrix is already in row-reduced form.
A matrix can be invertible without its RREF being explicitly written as $I_3$ before row operations.
Hence, this statement is false.
Step 4: Examine statement (C).
If $A$ has three distinct eigenvalues, then it must be diagonalizable and non-singular.
This guarantees that $A$ has full rank and is invertible.
As a result, its row-reduced echelon form must be the identity matrix $I_3$.
So, this statement is correct.
Step 5: Examine statement (D).
If the system $Ax=b$ has a solution for every possible $3 \times 1$ vector $b$, then $A$ must be invertible.
An invertible matrix always reduces to the identity matrix in row-reduced echelon form.
Therefore, this statement is also correct.
Step 6: Final conclusion.
Among the given statements, the only incorrect one is:
\[ \boxed{(B)} \]