The short answer is that the Thomas algorithm will be faster than any iterative scheme for almost all cases. The exception would perhaps be applying a single iteration of a very simple iterative scheme such as Gauss-Seidel, but this is highly unlikely to give an acceptable solution. Also, this is ignoring parallel processing concerns.
Multigrid is an especially poor choice in the case of a tri-diagonal matrix because although multigrid is $\mathcal O(n)$, the constant is quite large. In fact, multigrid does not even have an advantage over Gauss-Seidel until the matrices become quite large. This is due to the need for projection, prolongation, and relaxation operations for each multigrid level, each of which requires $\mathcal O(n)$ operations where n is the number of unknowns at that multigrid level.
Finally, this question is best addressed via operation counting. For the Thomas algorithm, a total of $5N$ multiplications and $3N$ additions are required for the solution. Iterative schemes require at least as many operations as matrix-vector multiplication and given a tri-diagonal matrix, each matrix-vector multiplication requires $3N-2$ multiplications and $2N-2$ additions. Therefore, even two applications of any (even the very simplest) iterative scheme will be more expensive than the Thomas algorithm.