Do explanatory variables have to have a linear relationship with the response variables? Do explanatory variables have to have a linear relationship with the response variable in multiple linear regression? What is the reason for this assumption?
Also, why are heteroscedastic relationships between IV's and DV's a problem in multiple regression?
 A: I assume you are talking about OLS/linear regression. Using OLS implies already that one assumes that there is a linear relationship. Why? Because you explain the response variable by a linear combination of the regressors. Hence, using OLS if you don't think that there is a linear relationship between the explanatory variables and the response variable defeats the purpose of OLS in the first place. Think about it like that: trying to identify a linear relationship between two variables when the true relationship isn't even close to linear is kind of like buying apples for a cherry pie.
If you are not talking about linear regression but non-linear regression, there is no assumption for a linear relationship between the response variable and the explanatory variables. Think about including the square of a regressor and calculating the partial effect for this regressor. The effect changes with the value of the regressor and hence, there is no linear relationship assumed or needed. Cheers.
A: The 'linear' in 'linear regression' means linear in the parameters, which isn't necessarily what people normally mean by 'linear' outside of statistics.  (To help clarify the issues, it may help you to read through this CV thread: How to tell the difference between linear and non-linear regression models?)  The linearity at issue isn't really an assumption, but just a statement of fact about the kind of model it is.
So is there, then, an assumption of the colloquial sense of linearity?  Sort of.  The model is being fit with the variables you chose and the structure / functional form you chose, and the results are conditional on those choices.  That said, you aren't required to simply input the raw form of each variable (call it '$X$'), you can input a transformation, $f(X)$, or several versions of it.  It is common, for example, to include both $X$ and $X^2$ in a regression model.  This works just fine.  The model really does just fit straight lines / flat planes / etc., but they can capture what you need.  To see this, it may help you to read my answer here: Why is polynomial regression considered a special case of multiple linear regression?
Regarding homoscedasticity, that is an assumption (see here).  The problem with using a model that assumes homoscedasticity with data that include substantial heteroscedasticity is that you are using the information in your data inefficiently to find the best parameter values and to understand the amount of noise in the data / uncertainty in the relationship.  That could mean that you have less power to detect an association (see here) or that you have an increased probability of type I errors (see here).
