Let $X$ be a discrete random variable (r.v.) taking distinct values $x_1,x_2,\dots$, and let $p_i:=P(X = x_i)$. Then the entropy of $X$ is defined by the formula
\begin{equation}
H(X): = \sum_i p_i \log \frac1{p_i}. \tag{1}
\end{equation}
Note that it does not matter whatsoever in what set/space the values $x_1,x_2,\dots$ are assumed to be; it does not matter if some of these values are close to, or far away from, one another -- in any sense.
What matters is that the $p_i$'s are the probabilities of distinct values of the r.v. $X$. Clearly, if we aggregate some of these values, then the entropy will go down; and if we split some of these values, then the entropy will go up.
Therefore, the formula $H(X) = \int dx\, p(x) \log \frac1{p(x)}$ will hardly make sense if, say, the integral is understood in the Riemann sense, implying the rather arbitrary grouping of the values $x$ according to the standard metric on $\mathbb R$. Moreover and much more importantly, the Riemann sums $\sum_i p(x_i)\Delta x_i \log \frac1{p(x_i)}\,$ for this integral are quite different from the sums $\sum_i p(x_i)\Delta x_i \log\frac1{p(x_i)\Delta x_i}$ that would genuinely correspond to the reality-based definition (1). Also, these latter sums will usually be very large if the $\Delta x_i$'s are very small, and the values of these sums may fluctuate wildly depending on the choice of the $\Delta x_i$'s.
The more general formula $H(X) = \int p(x) \log \frac1{p(x)}\, \mu(dx)$, where $\mu$ is a measure and the grouping of the values $x$ occurs according to the closeness of the corresponding values of $p(x)$ (!!), will hardly make more sense than the Riemann integral.
The only exception here would be when $\mu$ is the counting measure, with which no actual grouping (or splitting) of any values occurs. Then for the density (say $p$) of the distribution of the r.v. $X$ with respect to the counting measure $\mu$, the condition $\int p\,d\mu=1$ can be rewritten as $\sum_x p(x)=1$, which will imply that $p(x)\ne0$ only for (at most) countably many values of $x$, so that the r.v. $X$ is necessarily discrete -- and then we can write
\begin{equation*}
H(X) = \int p(x) \log \frac1{p(x)}\,\mu(dx)
=\sum_x p(x) \log \frac1{p(x)},
\end{equation*}
which is the same as (1), up to the change in notation.
So, if the r.v. $X$ is not discrete, then the only reasonable value to assign to the entropy of $X$ appears to be $\infty$, at least from the viewpoint of information theory.
As for the integral $\tilde H(X):=\int p(x) \log \frac1{p(x)}\, \mu(dx)$ in the case when $\mu$ is the Lebesgue measure, the main interest to it seems to be the easily seen fact (see e.g. Barron) that the maximum of $\tilde H(X)$ over all absolutely continuous r.v.'s $X$ with a fixed variance is attained when the distribution of $X$ is normal; moreover, the absolute value of difference of $\tilde H(X)$ from its maximum equals the relative entropy $\int p(x)\log\frac{p(x)}{\varphi(x)}\, dx$, where $\varphi$ is the normal density with the same mean and variance as $p$. However, what is actually used in the proofs is the relative entropy $\int \log\frac{dP}{dQ}\,dP$ (also known as the Kullback--Leibler divergence), which is well defined for any probability measures $P$ and $Q$ such that $P$ is absolutely continuous with respect to $Q$.
