Can a prime in a Dedekind domain be contained in the union of the other prime ideals?

Yes, it is possible.
According to Claborn's theorem¹ any abelian group is the class group of some Dedekind ring.
Take a Dedekind ring $R$ whose class group is isomorphic to $\mathbb Z$ and freely generated by the ideal $I$. Since $I=\mathfrak m_1 \mathfrak m_2\ldots \mathfrak m_N$ with all $\mathfrak m_i$'s maximal, one of those maximal ideals, call it $\mathfrak m$, must be without torsion. I claim that $\mathfrak m$ is contained in the union of the other maximal ideals of $R$.
Indeed, take an arbitrary nonzero $f\in \mathfrak m$ and decompose $(f)$ into a product of primes :
$$(f)=\mathfrak m^r.\prod \mathfrak n_i^{r_i}$$ ( $\mathfrak n_i\neq \mathfrak m, \quad $almost all $r_i=0$)
You can't have all the $r_i=0$, else $(f)=\mathfrak m^r$ would imply that $\mathfrak m$ is torsion in the class group.
Since $f$ is in all the maximal ideals $\mathfrak n_i$ with $r_i\neq0$ , the claim is proved : $\mathfrak m$ is contained in the union of the other maximal ideals of the Dedekind ring $R$.

An easy warm-up John (rightfully) evokes the prime avoidance theorem. It is easy to see that this theorem doesn't hold for infinitely many primes. For example consider the product ring $R=\mathbb Q^{\mathbb N}$ and the maximal ideals $\mathfrak m_n=\{(q_i)\in R | q_n=0\}\subset R$ . Then for the ideal $I=\mathbb Q^{(\mathbb N)}$ of almost zero sequences we have $I \subset \bigcup \mathfrak m_n$ although $I\nsubseteq \mathfrak m_n$ for each $n$.
This easy counterexample doesn't answer John's actual (more precise and more demanding) question .

Thanks to Jyrki who accurately pointed out (in a now tactfully deleted comment!) that my previous version incorrectly assumed that in Claborn's theorem I could take primes as free generators of the class group .

A mistake in a book (added later) In the book Algebraic Number Theory mentioned by John in his question the author describes (on page 66) a generalized localization. He starts with a completely general commutative ring $A$ and a completely arbitrary set $X\subset\text{Spec}(A)$ of prime ideals of $A$. He remarks that the complement $S=\text{Spec}(A)\setminus \bigcup \{\mathfrak p|\mathfrak p\in X\}$ is a multiplicative set and considers the ring of fractions $A(X)=S^{-1}A$. He writes that the only primes $\mathfrak q\subset A$ that survive in $A(X)$ are those which are subsets $\mathfrak q\subset \bigcup \{\mathfrak p|\mathfrak p\in X\}$, and this is absolutely correct. However he adds that in the case of a Dedekind ring $A$ the surviving ideals are those $\mathfrak p \in X$ . This claim (repeated page 70) is not true, as shown by taking for $A$ our $R$ above and for $X$ the set of all maximal ideals in $R$ different from $\mathfrak m$: that ideal $\mathfrak m$ survives in $R(X)$ although it is not an element of $X$ : $\mathfrak m \notin X $ by the very choice of $X$.
Congratulations to John for catching this very subtle little mistake made by a great arithmetician in a great book.

¹ C. R. Leedham-Green: The class group of Dedekind domains, Trans. Amer. Math. Soc. 163 (1972), 493-500 ; doi: 10.1090/S0002-9947-1972-0292806-4, jstor.

If $R$ is the ring of integers $O_K$ of a finite extension $K$ of $\mathbf{Q}$, then I don't think this can happen. The class of the prime ideal $P$ is of finite order in the class group, say $n$. This means that the ideal $P^n$ is principal. Let $\alpha$ be a generator of $P^n$. Then $\alpha$ doesn't belong to any prime ideal other than $P$, because at the level of ideals inclusion implies (reverse) divisibility, and the factorization of ideals is unique.

This argument works for all the rings, where we have a finite class group, but I'm too ignorant to comment, how much ground this covers :-(

This answer refers to the contributions of Jyrki and Georges: assume that a maximal ideal $P$ of a Dedekind domain $R$ is NOT contained in the union of all other maximal ideals. Then there exists an element $f\in P$ such that $v_P(f)=n>0$ for the discrete valuation attached to $P$ and $v_Q(f)=0$ for all $Q\neq P$. Now $P^n$ consists of those elements $r\in R$ such that $v_P(r) \geq n$. Thus for every $r\in P$ we get $r=fs$ with $s\in R$. Hence $P^n =fR$. Jyrki has already shown that if $R$ has torsion class group, then no maximal ideal contained in the union of all others can exist.

So: a maximal ideal of $R$ contained in the union of all other maximal ideals exists if and only if the class group of $R$ is not torsion.