MA(q) model i.i.d assumption An MA(q) model is 
$X_t = \mu + \sum_{j=1}^q{a_j \epsilon_{t-j}} + \epsilon_t$
where $\{\epsilon_t\} \sim WN(0, \sigma^2)$
Is it necessary for $\{\epsilon_t\}$ to be i.i.d white noise? My guess is that it is not necessary because one example of an MA(q) model is the k-period long return
$r_t[k] = r_t + r_{t-1} + r_{t-2} + ... + r_{t-k}$
In this case, $a_j = 1$, $\mu = 0$ and $r_t = \epsilon_t$. I don't think that the log return $\{r_t\}$ are i.i.d.
 A: It is not a model assumption, it is its definition.
Of course, you can define another model where the shocks are non-i.i.d., but you will have to give another name to that model; it will no longer be MA(q) as we know it.
A: Different authors may have different definitions, but in the wide sense it is not necessary for the $\varepsilon_t$ in an MA(q) to be independent; they are simply required to be uncorrelated, with an identical, unspecified marginal distribution. Some authors do restrict the definition of MA(q) to have independent $\varepsilon_t$ and for each to have a normal distribution, however (to derive estimators, test statistic distributions, etc).
Concerning your last point: log-returns of financial assets are indeed frequently modeled as being uncorrelated, or even independent. Think of, for example, the Black-Scholes model. Depending on the goal for modelling, it may not be possible to build a better-performing model. Intuitively, you would not expect any easy-to-model dependence structure over time to exist; if there was, it would tend to be taken advantage of, and then disappear.
