It turns out that this is a surprisingly deep question.
Correlation
First consider the difference between the absence of correlation between two variables (e.g. Pearson correlation of 0) and statistical independence. Below are a number of examples where the correlation is 0, but the variables are not independent. The value of X definitely provides information about the value of Y. (from Wikipedia: Correlation and Dependence)
A broad range of causal relationships might underlie these observations.
Statistical Dependence vs. Causation
Assuming you meant that you are looking for examples of statistical dependence without causation, the answer is typically (and perhaps surprisingly) NO (well, see edit “*” below). Asymptotically, statistical dependence between two variables (contrary to common wisdom) DOES imply causation, just not necessarily directed from one of the two variables to the other (see below, Storm v Mercury). For example, they may both be caused by a third variable (I think Emily Glassberg Sands‘s answer gets at this).
Causation vs. Statistical Dependence
On the other hand, as Michael Lamar points out, the inverse is not true. Statistical independence only allows non-causal relationships, it doesn’t require them. There can be causality without statistical dependence. I think this is the level at which you are asking the question.
Causation without Evidence
There are other examples where causation exists without observable evidence, so it cannot be detected. For example, X causes Y, but X can only be observed in one state. At some point X may change - causing a change in Y, but it may be impossible to identify this causation from available data. Henning Strandin’s answer touches on this (I sought this as my PhD topic, too, but couldn’t get ready support at the time, you go Henning!).
Actual vs. Statistical Causation
Beyond statistical causation (e.g. the tendency of a dropped match X to cause a fire Y), there is the question of actual causation (did a particular dropped match cause a particular fire). While this seems intuitively simple, formalizing this rigorously is surprisingly difficult. Some inroads have been made in the last 20 years and here is one example I am familiar with (Joseph Halpern’s , Actual Causality).
Statistical Causation
Causation is of particular interest because knowledge of causation provides a means to effect change. This is what many folks are looking for and can be absent even from highly accurate predictive models. However, identifying causation from observations without experimentation involves a process called counterfactual reasoning, which introduces a potentially unobservable hypothetical (what would have happened IF I had done this, instead of what I actually did). Judea Pearl (computer scientist) (or here) formalized this concept of action in a statistical setting via the “do()” calculus in his book Causality. A key insight is that intervening to set X (do(X=1)) is very different from a mere passive observation of X=1.
(Thanks to Keyon Vafa for this image)
For an intuitive example, consider the difference between observing a car driving in first gear, as opposed to manually forcing a car into first gear while driving without changing any of the other operating conditions.
Ouch.
With apologies, I’ve spent far too long on this answer. Please message me if you have interests in this area.
UPDATE 2023-MAY - correlation based on sample bias / without causation
While rereading Pearl’s Book of Why, I came across a great example showing correlation arising from sampling bias. Consider two independent coins ({H,T} - binary outcomes), where samples are only recorded when at least one is heads. In this sample, whenever one coin appears as tails, the other appears as heads, a clear correlation — but without causation, just sample bias. A clear picture from another Quora Posting