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I have a counterexample from potential theory. Let $\{x_{k}|k\geq 1\}$ be a countable dense subset of $[0,1]$ and let $\sum_{k=1}^{\infty}a_{k}<\infty$. Let $f(z)=\sum_{k=1}^{\infty}a_{k}\mathrm{Log}(x-x_{k})$, and let $E=f^{-1}[\{-\infty\}]$. Let $E_{n}=f^{-1}[\infty,-n)\cap[0,1]$. Then each $E_{n}$ is a dense open subset of $[0,1]$. However, $E=\bigcap_{n}E_{n}$ has Hausdorff dimension zero. In fact, in potential theory we say that $E$ is a polar set. I suspect that one can get many other counterexamples which are much "thinner" by replacing Log with a function that has a much thinner singularity.