Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($ab=0$) idempotents in $R$. I am looking for a characterization for rings with the property that if $e$ is a primitive idempotent of $R$ and $e=r+s$, where $r, s\in R$, then $(e)\subseteq (r)$ or $(e)\subseteq (s)$ where $(x)$ is the ideal generated by $x$ in $R$.

The property you are looking at is that of a local idempotent.

Let $R$ be a commutative ring with $1$. A *local idempotent*
of $R$ is an idempotent $e\in R$ such that $eRe = eR$ is a local ring.
Local idempotents are always primitive, but primitive idempotents
need not be local (e.g., $1\in\mathbb Z$ is primitive but not local).

Here is the argument that your property is the same as being local.

Assume that $e$ is a local idempotent. Then $R\cong eR\times (1-e)R$, where the factor $eR$ is a local ring. In coordinates, $e = (1,0)$. Suppose that $e=r+s$. Writing this in coordinates yields $(1,0)=(r_1,r_2) + (s_1,s_2)$. Since $r_1, s_1\in eR$, which is local, one of $r_1$ or $s_1$ is a unit in $eR$. Suppose that it is $r_1$. Then $e = (1,0) = (r_1,r_2)*(r_1^{-1},0) \in (r)$, or $(e)\subseteq (r)$. Similarly, if $s_1$ is a unit in $eR$, then $e\in (s)$, or $(e)\subseteq (s)$.

Now suppose that $e$ is not local, so $eR$ is not a local ring. Let $M, N\subseteq eR$ be distinct maximal ideals. Then $M+N=eR$, so there exist $m\in M$ and $n\in N$ such that $m+n=1$ in $eR$. Thus, writing in coordinates in $R \cong eR\times (1-e)R$, we have $(m,0) + (n,0) = (1,0) = e$. I let $r = (m,0)$ and $s=(n,0)$, so $r+s=e$. But $e=(1,0)\notin (r)$, since $(r) \subseteq M\times \{0\}$ and $(1,0)\notin M\times \{0\}$. Similarly $e\notin (s)$.

You have stated ``I am looking for a characterization for rings with the property that if $\ldots$ ETC.'' One (partial?) answer is: those rings whose primitive idempotents are local. I'm not sure how much more you can say than this, but the following paper

W.K. Nicholson.

I-rings.

Trans. Amer. Math. Soc. 207 (1975), 361--373.

proves that primitive idempotents are local in any ``semipotent'' ring. A commutative ring is semipotent if every ideal not contained in the Jacobson radical contains a nonzero idempotent.