For creating the joint posterior distribution for multiple variables, are the associated Bayesian priors usually assume independent of each other? Suppose that we have data, $D$, and two parameters we want to learn about, $\theta_1, \theta_2$. We will usually put priors on $\theta_1, \theta_2$, then have the expression:
$$
p(\theta_1, \theta_2\mid D) \propto p(D\mid \theta_1, \theta_2) p(\theta_1) p(\theta_2)
$$
I am wondering why most set-ups assume that the priors above are independent. What happens if we do not have it?
 A: Your expression is only correct when you assume independent priors. Otherwise, the expression would become
$$
p(\theta_1, \theta_2 | D) \propto p(D|\theta_1, \theta_2) p(\theta_1, \theta_2)
$$
In this expression, you may need further assumptions to work with $p(\theta_1, \theta_2)$ or you can work with it as is. If you assume independence you get your expression again. From what I've seen, it's also common to factor this expression as
$$
p(\theta_1, \theta_2) = p(\theta_1|\theta_2)p(\theta_2)
$$
This requires no further independence assumptions, and it might be easy to specify the conditional distribution $p(\theta_1|\theta_2)$. Having said that, in many cases it can be completely justifiable to have independent priors on $\theta_1$ and $\theta_2$ so it really depends on the individual problem you're trying to solve.
A: As Maurits M mentions in another answer whether independence makes sense is really problem specific. The OP question asked: 
"I am wondering why most set-ups assume that the priors above are independent. What happens if we do not have it?"
which is really 2 questions. 
(1) Why most set-ups assume the priors are independent? 
My guess (w/o being able to read minds) would be that multivariate distributions that can be written in closed form are few and far between. 
This is also the reason why MCMC techniques are so popular. It is much easier to write a product of marginal priors and may make the sampler easier to write down. 
(2) What happens if we do not have it?
This question could be interpreted as the impact of incorrectly specifying independence, or as to how to proceed if you know independence is not a reasonable assumption. I'll answer both.  
If you've incorrectly assumed independence, then the degree of the impact will depend on how egregious this violation is w.r.t. the true underlying model. For example, naive Bayes assumes independence and often works well even if the independence is empirically not true. The reason is that often symmetries exist in the data generation mechanism which "cancel out" the independence violations. However, this statement is more of an empirical justification and I'm unaware of any group theoretic proof of this claim. 
