If A causes B which causes C, is there omitted variable bias when regressing B on C without including A? If A causes B which causes C, is there omitted variable bias when regressing B on C without including A?
Let's say I know the hot weather causes (or at least tends to cause) people to order larger quantities of ice cream (per order).  I want to know the effect of 1 scoop of ice cream on service time (per order) at an ice cream shop.  
A <- The weather temp
B <- Quantity of Ice Cream Ordered (Per Order)
C <- Service Time (Per Order)
If I regress B on C, am I biasedly omitting A?  My thinking is there probably will be a correlation between A and C due to its relationship with B.  But the weather does not realistically have any effect on how fast the workers scoop ice cream.  That is my basic question about the second omitted variable bias assumption.  
 A: Technically no. You must have a direct effect relating A to C conditional on B to observe a change in outcome. Adjusting for factors (in this case A) which are independent of the outcome (C) conditional on other factors in the model (B) will not change the estimate or standard error of the main association (B to C).
If A predicts C and is independent of B, then adjustment for A will lead to more efficient inference, by reducing the standard deviation of the residuals and thus shrinking the confidence intervals. If A predicts (causally) both B and C, with the latter causation being conditional on B, then A is a confounder and omitting the variable introduces bias.
A: 
But the weather does not realistically have any effect on how fast the workers scoop ice cream.

It's not inconceivable that it could!
One approach to your problem would be to use principal components, which take the correlation between B and A into account. The largest component will likely be directly related to B (and most of A). Then, if the effect of the second component on C has no statistical significance, you can remove it.
A: This is related to the causal Markov condition, which is important in the causal graph literature.  
In the context of your setup, the causal Markov condition (plus two assumptions*) implies that $A$ and $C$ are independent conditional on $B$:  $p(C|A,B) = p(C|B)$.  In other words, if the causal Markov condition is satisfied, then knowing the temperature and order quantity doesn't tell us anything more about serving time than knowing the order quantity alone.  If this condition is satisfied, then there is no omitted variable bias if you regress $C$ on $B$ and not $A$.  
[*] The two assumptions are (1) that $A$ does not have any direct effect on $C$; instead any effect of $A$ on $C$ is mediated by other variables; and (2) that $B$ is the only variable that directly effects $C$.  If (1) is false, then $A$ and $C$ are not necessarily independent conditional on $B$.  If (1) is true but (2) is false, then $A$ and $C$ are independent only conditional on all of the variables that directly effect $C$.  
A: 
If A causes B which causes C, is there omitted variable bias when regressing B on C without including A?

If A causes B causes C, but A does not cause C except via B, then leaving out A will not affect your ability to identify B's effect on C.  
If, on the other hand, A causes C both directly and by affecting B then you won't identify the effect of B on C if you leave out A, so there will be omitted variable bias.

If I regress B on C, am I biasedly omitting A? My thinking is there probably will be a correlation between A and C due to its relationship with B.

There will indeed be such a correlation, but that may or may not be relevant to the question of B's effect on C.
If we assume that A affects C only through B, then if you regress C on A you will identify the effect of A on C, which is not what you want. If you regress C on B you will identify the effect of B on C, which is what you want. And if you regress C on A and B then you will still identify the effect of B on C and the coefficient on A will be 0 (up to sampling error).  In the terminology: A and C are conditionally independent given B.
DAGitty is quite useful for answering these sorts of questions in more complicated scenarios.  Draw out your graph and see what, if anything, needs to be conditioned on to get what you want.
