Is it known whether $BPP\cap NP\subseteq RP$?
As with most questions in complexity, I'm not sure there will be a full answer for a very long time. But we can at least show that the answer is non-relativizing: there is an oracle relative to which inequality holds and one relative to which equality holds. It's fairly easy to give an oracle relative to which the classes are equal: any oracle which has $\mathrm{BPP} = \mathrm{RP}$ will work (eg any oracle relative to which "randomness doesn't help much"), as will any oracle which has $\mathrm{NP} \subseteq \mathrm{BPP}$ (eg any oracle relative to which "randomness helps a lot"). There are a lot of these, so I won't bother with the specifics.

It's somewhat more challenging, though still fairly straightforward, to design an oracle relative to which we get $\mathrm{RP} \subsetneq \mathrm{BPP} \cap \mathrm{NP}$. The construction below actually does a bit better: for any constant $c$, there is an oracle relative to which there is a language in $\mathrm{coRP} \cap \mathrm{UP}$ which is not in $\mathrm{RPTIME}[2^{n^c}]$. I'll outline it below.

We'll design an oracle $A$ that contains strings of the form $(x,b,z)$, where $x$ is an $n$-bit string, $b$ is a single bit, and $z$ is a bit string of length $2n^c$. We will also give a language $L^A$ which will be decided by a $\mathrm{coRP}$ machine and a $\mathrm{UP}$ machine as follows:


The $\mathrm{coRP}$ machine, on input $x$, guesses $z$ of length $2|x|^c$ randomly, queries $(x,\mathtt{0},z)$, and copies the answer.
The $\mathrm{UP}$ machine, on input $x$, guesses $z$ of length $2|x|^c$, queries $(x,\mathtt{1},z)$, and copies the answer.


To make the above-specified machines actually meet their promises, we need $A$ to satisfy some properties. For every $x$, one of these two options must be the case:


Option 1: At most half of $z$ choices have $(x,\mathtt{0},z) \in A$ and zero $z$ choices have $(x,\mathtt{1},z) \in A$. (In this case, $x \not\in L^A$.)
Option 2: Every $z$ choice has $(x,\mathtt{0},z) \in A$ and precisely one $z$ choice has $(x,\mathtt{1},z) \in A$. (In this case, $x \in L^A$.)


Our aim will be to specify $A$ satisfying these promises so that $L^A$ diagonalizes against every $\mathrm{RPTIME}[2^{n^c}]$ machine. To try to keep this already long answer short, I'll drop the oracle construction machinery and a lot of the unimportant details, and explain how to diagonalize against a particular machine. Fix $M$ a randomized Turing machine, and let $x$ be an input so that we have full control over the selection of $b$'s and $z$'s so that $(x,b,z) \in A$. We will break $M$ on $x$.


Case 1: Suppose there is a way to select the $z$'s so that $A$ satisfies the first option of its promise, and $M$ has a choice of randomness which accepts. Then we will commit $A$ to this selection. Then $M$ cannot simultaneously satisfy the $\mathrm{RP}$ promise and reject $x$. Nevertheless, $x \not\in L^A$. So we have diagonalized against $M$.
Case 2: Next, assume that the previous case did not work out. We will now show that then $M$ can be forced either to break the $\mathrm{RP}$ promise or to reject on some choice of $A$ satisfying the second option of its promise. This diagonalizes against $M$. We will do this in two steps:


Show that for every fixed choice $r$ of $M$'s random bits, $M$ must reject when all of its queries of the form $(x,\mathtt{0},z)$ are in $A$ and all of its queries of the form $(x,\mathtt{1},z)$ are not in $A$.
Show that we can flip an answer $(x,\mathtt{1},z)$ of $A$ for some choice of $z$ without affecting the acceptance probability of $M$ by much.


Indeed, if we start with $A$ from step 1, $M$'s acceptance probability is zero. $A$ doesn't quite satisfy the second option of its promise, but we can then flip a single bit as in step 2 and it will. Since flipping the bit causes $M$'s acceptance probability to stay near zero, it follows that $M$ cannot simultaneously accept $x$ and satisfy the $\mathrm{RP}$ promise.


It remains to argue the two steps in Case 2:


Fix a choice of random bits $r$ for $M$. Now simulate $M$ using $r$ as the randomness and answering the queries so that $(x,\mathtt{0},z) \in A$ and $(x,\mathtt{1},z) \not\in A$. Observe that $M$ makes at most $2^{n^c}$ queries. Since there are $2^{2n^c}$ choices of $z$, we can fix the unqueried choices of $z$ to have $(x,\mathtt{0},z) \not\in A$, and have $A$ still satisfy the first option of its promise. Since we couldn't make Case 2 work for $M$, this means $M$ must reject on all its choices of randomness relative to $A$, and in particular on $r$. It follows that if we select $A$ to have $(x,\mathtt{0},z) \in A$ and $(x,\mathtt{1},z) \not\in A$ for every choice of $z$, then for every choice of random bits $r$, $M$ rejects relative to $A$.
Suppose that for every $z$, the fraction of random bits for which $M$ queries $(x,\mathtt{1},z)$ is at least $1/2$. Then the total number of queries is at least $2^{2n^c} 2^{2^{n^c}}/2$. On the other hand, $M$ makes at most $2^{2^{n^c}} 2^{n^c}$ queries across all its branches, a contradiction. Hence there is a choice of $z$ so that the fraction of random bits for which $M$ queries $(x,\mathtt{1},z)$ is less than 1/2. Flipping the value of $A$ on this string therefore affects the acceptance probability of $M$ by less than $1/2$.