Does no correlation imply no causality? I know that correlation does not imply causality but does an absence of correlation imply absence of causality?
 A: 
does an absence of correlation imply absence of causality?

No. Any controlled system is a counterexample.
Without causal relationships control is clearly impossible, but successful control means - roughly speaking - that some quantity is being maintained constant, which implies it won't be correlated with anything, including whatever things are causing it to be constant.
So in this situation, concluding no causal relationship from lack of correlation would be a mistake.
Here's a somewhat topical example.
A: The better answer to the question is that correlation is a statistical, mathematical, and/or physical relationship while causation is a metaphysical relationship. You can't LOGICALLY get from correlation (or non-correlation) to causation, without a (large) set of assumptions binding the metaphysics to the physics. (One example is that what two people might agree to be "a rational observer" is to a large degree arbitrary and probably ambiguous). If A pays B to do C which results in D, what is D's cause? There's simply no rational reason to choose C or B or A (or any of A's precursor events).
Control theory deals with systems in realms where they are under control. One way to get a dependent variable under control is to reduce the response of that variable to the possible range of (controlled) variation of the independent variable to statistical noise. For instance, we know air pressure correlates to health (just try breathing vacuum), but if we control air pressure to 1 +/-0.001 atm, how likely is ANY variation of air pressure to effect health?
A: Yes, contrary to previous replies. I'm going to take the question as nontechnical, particularly the definition of "correlation". Maybe I'm using it too broadly, but see my second bullet.  I hope it will be considered appropriate to discuss other answers here, because they illuminate different portions of the question. I'm drawing on Pearl's approach to causation, and in particular my take on it in some papers with Kevin Korb.  Woodward probably has the clearest nontechnical account. 


*

*@conjugateprior says "any controlled system is a counterexample". Yes, to the stronger claim that noncorrelation observed in your experiment implies no causation. I'm going to assume the question is more general. Certainly one experiment might have failed to control for masking causes, or inappropriately controlled for common effects, and hidden the correlation. But if $x$ causes $y$, there will be a controlled experiment where that relationship is revealed. Almost all definitions or accounts of causation treat it as a difference that makes a difference.  Therefore no causation without (some kind of) correlation. If there is a direct link $x \rightarrow y$ in a causal Bayesian network, it does not mean that $x$ always makes a difference to $y$, only that there is some experiment fixing all other causes of $y$ where wiggling $x$ wiggles $y$.

*@aksakal has a great example why linear causation is insufficient. Agreed, but I want to be broad and nontechnical. If $y=x^2$, it's  incomplete to tell a client that $y$ is uncorrelated with $x$. So I'll use correlation very broadly to mean a difference in $x$ that is reliably associated with a difference in $y$. It can be as nonlinear or nonparametric as you like. Threshold effects are fine ($x$ makes a difference to $y$, but only over a finite range, or only by being larger or smaller than a particular value, like voltage in digital circuits).

*@Kodiologist creates an example where $y = \mathrm{Unif}({x,-x})$, so $|y| = |x|$ but no linear correlation. But clearly there is a discoverable relationship, so correlated in the broad sense.

*@Szabolcs uses random number generators to show an output stream constructed to appear uncorrelated. Like the digits of $\pi$, the stream appears random but is deterministic. I agree you're unlikely to find the relationship if given only the data, but it's there. 

*@Li Zhi notes you can't logically jump from correlation to causation. Yes, no causes in, no causes out.  But the question begins from causation: does it imply correlation?  In the air pressure example, we have a threshold effect. There is a range where air pressure is uncorrelated with health. Indeed plausibly where it has no causal effect on health. But there is a range where it does. That is sufficient. But probably better to note ranges where there is and is not an effect.  If $A \rightarrow B \rightarrow C \rightarrow D$, then there is correlation all along the chain, because there is causation.  Repeated observation (or experiment) can show that $A$ does not directly cause $D$ but the correlation is there because there is a causal story.
I do not know what @user2088176 had in mind, but I think if we take the question very generally, then the answer is yes. At least I think that's the answer required of the causal discovery literature and the interventionist account of causation.  Causes are differences that make a difference. And that difference will be revealed, in some experiment, as persistent association. 
A: No. Mainly because by correlation you most likely mean linear correlation. Two variables can be correlated nonlinearly, and may show no linear correlation. It's easy to construct an example like that, but I'll give you an example which is closer to your (narrower) question.
Let's look at the random variable $x$, and the non random function $f(x)=x^2$, with which we create a random variable $y=f(x)$. The latter is clearly caused by the former variable, not just correlated. Let's draw a scatter plot:

Nice, clear nonlinear correlation picture, but in this case it's also direct causality. However, the linear correlation coefficient is non significant, i.e. there's no linear correlation despite obvious nonlinear correlation, and even causality:
>> x=randn(100,1);
>> y=x.^2;
>> scatter(x,y)
>> [rho,pval]=corr(x,y)

rho =

    0.0140


pval =

    0.8904

UPDATE:
@Kodiologist is right in the comment. It can be shown mathematically that linear correlation coefficient for these two variables is zero indeed. In my example $x$ is the standard normal variable, so we have the following:
$$E[x]=0$$
$$E[x^2]=1$$
$$E[x\cdot x^2]=E[x^3]=0$$
Hence, the covariance (and subsequently the correlation) is zero:
$$Cov[x,x^2]=E[x \cdot x^2]-E[x]E[x^2]=0$$
We'd get the same result for any symmetrical distribution, such as uniform $U[-1,1]$.
A: No. In particular, random variables can be dependent but uncorrelated.
Here's an example. Suppose I have a machine that takes a single input $x ∈ [-1, 1]$ and produces a random number $Y$, which is equal to either $x$ or $-x$ with equal probability. Clearly $x$ causes $Y$. Now let $X$ be a random variable uniformly distributed on $[-1, 1]$ and select $Y$ with $x = X$, inducing a joint distribution on $(X, Y)$. $X$ and $Y$ are dependent, since
$$
P(X < -\tfrac{1}{2})P(|Y| < \tfrac{1}{2}) = \tfrac{1}{4} \cdot \tfrac{1}{2} = \tfrac{1}{8} ≠ 0 = P(X < -\tfrac{1}{2},\; |Y| < \tfrac{1}{2}).
$$
However, the correlation of $X$ and $Y$ is 0, because
$$
\operatorname{Corr}(X, Y)
= \frac{\operatorname{Cov}(X, Y)}{σ_Xσ_Y}
= \frac{E[XY] - E[X]E[Y]}{σ_Xσ_Y}
= \frac{0 - 0\cdot0}{σ_Xσ_Y}
= 0.
$$
A: Maybe looking at it from a computational perspective will help.
As a concrete example, take a pseudorandom number generator.
Is there a causal relationship between the seed you set and the $k^\text{th}$ output from the generator?
Is there any measurable correlation?
