Let's consider a uniformly distributed random variable: $\theta \sim U(-\frac{1}{2}, \frac{1}{2})$. Then evidently,
$$E[\theta]= \int_{-\infty}^{\infty} \theta \, p(\theta) \, d\theta= \int_{-\frac{1}{2}}^{\frac{1}{2}} \theta \, d\theta= 0,$$
and thus,
$$\frac{E[\theta]}{1 - E[\theta]} = 0.$$
On the other hand,
$$E\left[\frac{\theta}{1 - \theta}\right]= \int_{-\infty}^{\infty} \frac{\theta}{1 - \theta} \, p(\theta) d\theta= \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{\theta}{1 - \theta} \,d\theta= \bigg[ -\theta - \ln |1-\theta| \bigg]_{-\frac{1}{2}}^{\frac{1}{2}} = \ln 3 - 1.$$
And so for this example,
$$\frac{E[\theta]}{1 - E[\theta]} \neq E\left[\frac{\theta}{1 - \theta}\right].$$
Similar derivations for $\theta \sim \text{Beta}(5, 3)$ are an exercise left to the reader.
