OK, using the modern (Lorentz invariant) definition of mass, the answer is that the center of mass does not change due to relalativisitc effects, but can be expected to change as the bar stretches under tension.
Let's tackle those two statements one at a time.
Relativistic effects.
Mass defined this way is unaffected by relative velocity
The distance from the center of any given mass element is unaffected by the spinning because that distance is always at right angles to it's velocity relative the center.
Simple mechanical stress
That said, any given portion of the bar must supply the centripetal force to hold the outlying parts of the bar in their path, and will accordingly stretch a little (how much depends on the materials Young's modulus, the geometry and density of the bar and the speed of rotation. These are the ordinary mechanical considerations that set the operational limit for smoothly turning machines.
What you might have wanted to ask and why you shouldn't
You might have had in mind a definition of "relativistic mass", that has mass increasing with speed. That notion comes from a decision about how to write the expression for a moving particle's total energy:
$$ E = \gamma m c^2 = \frac{mc^2}{\sqrt{1 - (v/c)^2}} \,.$$
In the old way of thinking and talking about these things the symbol $m$ would be called "rest mass" and the combination $\gamma m$ was called "relativistic mass", which let you keep a few equations in the same form they had in Newtonian mechanics. (In the modern parlance $m$ is the only mass or sometimes the "invariant mass" when we want to be clear, and $\gamma m$ doesn't get a name at all.)
However, the idea that "mass changes as speed changes" leads people astray: it lets them form linguistic chains of reasoning that are superficially reasonable but break the fundamental rules of relativity, resulting is questions like "If I throw a baseball hard enough can it's mass be so large it forms a black whole?". That sounds reasonable if you think that mass is a function of velocity, but it is emphatically not reasonable, because in the ball's frame of reference it is at rest and nothing has changed. Using the invariant mass makes that question impossible and improves your reasoning about relativity.
How you could still ask the "relativistic mass" version of the question
We can simply define the "center of energy" as the first moment of energy in the same way that mass is defined as the first moment of mass. (Don't be put off by the old fashioned language, I mean put an $E$ where ever you see $m$ in the formula you are used to.) If you do that then the answer is "Yes, the center of energy changes as the bar picks up speed".
