What is the implication of unit root of MA?

A ARMA(p,q) process is weakly stationary, iff the root of its AR part is not on the unit circle. So its weak stationarity doesn't depend on its MA part. But what can the positions of the roots of its MA part imply?

In the unit root tests for ARIMA, a unit root of the MA polynomial indicates that the data were overdifferenced. Does it mean that the differenced time series is not weakly stationary? If yes, does it contradict to the earlier fact that the weak stationarity of ARMA doesn't depend on its MA part?

If the roots of the MA process indicate a violation this can be due to a variety of causes;

1. Over-differencing of Y
2. Redundancy of the AR and MA structure
3. Omitted deterministic variables ( Pulses/Level Shifts/Seasonal Pulses/Local time trends
4. Incorrect Power Transformation
5. Changes in parameters over time
6. Changes in error variance over time
7. Omission of user-specified causal variables

Hope this helps ...why model identification is not " a walk in the woods " and shouldn't be accomplished using simp[le-minded AIC/BIC tests but rather aggressively/comprehensively formulated.

To elaborate on some of the above points, consider differencing a process following a deterministic trend $y_t=a+bt+\epsilon_t$.

$\Delta y_t$ is not invertible, as $\Delta y_t=bt-b(t-1)+\Delta\epsilon_t=b+\Delta\epsilon_t$. This is an $MA(1)$ with a unit root, and hence not invertible. This is because first differencing is the "wrong" detrending scheme for a trend stationary process.

Also, we have that the long-run variance of an $MA(1)$ process can be written as $$J=\sigma^2(1+\theta)^2,$$ as \begin{eqnarray*} J &=& \sum_{j=-\infty}^{\infty}\gamma_j \\ &=& \gamma_0 + 2 \sum_{j=1}^{\infty}\gamma_j \\ &=& \gamma_0 + 2 \gamma_1 + 0\\ &=& \sigma^2(1 + \theta^2) + 2 \theta \sigma^2\\ &=& \sigma^2(1 + \theta^2 + 2\theta)\\ &=& \sigma^2(1 + \theta)^2 \end{eqnarray*} We have $J=0$ for $\theta=-1$, so an $MA(1)$ with a unit root. This is a problem for example because the long-run variance is asymptotic variance of the sample mean, $$\sqrt{T}(\bar{Y}_T-\mu)\to_d N\Biggl(0,\sum_{j=-\infty}^{\infty}\gamma_j\Biggr),$$ which is for instance used for standard errors - which should not be zero.

I think if you are sure that the process is ARMA, then the MA part doesn't affect the stationarity. But if you are not sure of that, unit root tests of the MA part may suggest that it's "likely" that the process as specified is not actually ARMA (and so you would want to integrate it).