Heteroskedasticity and Type I, Type II Error If one is conducting a hypothesis test for heteroskedasticity, would one generally consider a Type I or a Type II error more serious? What is the reasoning behind this?
 A: If you reject homoscedasticity and opt for using robust standard errors instead when homoscedasticity is actually present, you lose efficiency and finite sample properties. Asymptotically however your model inference will be still be valid. 
However, if you opt for non robust standard error, and heteroscedasticity is present, your inference is always plain wrong. 
If you do applied research (or industry work) you should clearly always use robust errors, you lose nothing if your wrong - provided your sample is large - whereas the other way around you loose the ability to do valid inference. 
A: The answer to the question is contained in @Glen_b 's initial comment:  To know if a type I or type II error is more serious, you have to know the consequences of each.
Is it better to be late to a party or early to a party?  Well, the Red Queen imprisons all those who show up late for her parties, and the White Queen imprisons all those who show up early for her parties.  So, it depends on whose party you are going to.
The OP doesn't give any context to why this hypothesis test is being conducted.
But imagine you are conducting an initial screening trial of several treatments, with only a few observations for each treatment.  Here, our hypothesis tests are likely to have low power, and we want to be sure to retain several treatments for future testing. In this case, we are likely to be more tolerant of type I errors, and want to avoid type II errors.  False positives can be weeded out later.  But we don't want to miss a potentially-useful treatment by making a false-negative error.
For an opposing case, if we are comparing a new medical treatment to an established one, we will likely want to accept a type-II error over a type-I error.  A false positive will cost lives and money.  A false negative may mean we miss out on a new potentially-important treatment.  But what's lost with the false negative is only the marginal improvement of the new treatment relative to the established one.
Obviously, there are other considerations that come in to play in these kinds of decisions.
As a side note, the answer by @Repmat suggests that the question is posed relative to testing for homogeneity of variance as a precedent to conducting another analysis, say, a t-test or anova.  The OP doesn't indicate that this is in the context of the question.  But if it is, it's worth pointing out that it is generally not recommended to determine the type of test conducted (e.g. Student's t-test vs. Welch's t-test) contingent on the results of another hypothesis test (e.g. a test for homogeneity).

Source:  I've used similar examples here.
