Is expectation Riemann-/Lebesgue–Stieltjes integral?

If $f$ is continuous, then $\int_a^b {f(x)dF(x)} = \int_a^b {f(x)d\mu (x)} $, for any $-\infty < a < b < \infty$, where $F$ and $\mu$ are the distribution function and probability distribution of $X$, respectively (they are related by $\mu((s,t])=F(t)-F(s)$, for any $-\infty < s < t < \infty$).

A drawback of the Riemann-Stieltjes integral is illustrated in the following simple example. Suppose that $X=0$ almost surely. Then, $\int {F(x)dF(x)} $ is not defined, whereas $\int {F(x)d\mu (x)} = F(0) = 1$. Of course, ${\rm E}[F(X)] = {\rm E}[F(0)]= 1$.

Another (more significant) drawback is indicated in GWu's answer.

This question is closely related to your other question regarding Riemann-Stieltjes integrals.

In the case that $f$ is continuous, these two types of integral agree provided they are finite.

But there are also other cases. For instance, $f$ is merely Borel measurable, then the Lebesgue-Stieltjes integral $\int f(x) dg(x)$ is defined, but not Riemann-Stieltjes integral because $f$ might be unbounded. This still have probabilistic interpretation because in this case $f(X)$ is still a random variable, and we can still consider the expectation of $f(X)$.

It maybe better for you to use the notation $$ \mathsf{E}[f(X)] = \int\limits_{\mathbb{R}}f(x)Q(dx) $$ where $Q$ is a distribution of an r.v. $X$. Then you do not need to worry if $X$ is a continuous r.v. or a discrete one. Then depending on what you know about $X$ (sumulative distribution function $g$ or a density function $h$) you will rewrite the first equation using $$ Q(dx) = dg(x) = h(x)\,dx. $$ Usually only Lebesgue (Lebesgue-Stieltjes) integrals are used in the probability theory. On the other hand to calculate them you can use an equivalence of Lebesgue-Stieltjes and Riemann-Stieltjes integrals (provided necessary conditions).

Edited: For the discrete distribution CDF $g$ is a pure jump function with jumps at values $\alpha_i$ of the r.v. $X$ and sizes of jumps $p_i$ such that $p_i = \mathsf{P}(X = \alpha_i)$. Then the integral is again a Lebesgue integral above can be rewritten as a Lebesgue-Stieltjes integral using $g$ or as a sum.