I prove your question for (not necessarily Noetherian) commutative ring. Irreducible ideals in non-Noetherian ring are complicate (see this question). For Noetherian ring, see our paper for a study of the index of reducibility. The proof is elementary but quite long.
We can assume that $I = 0$. Suppose $ 0 \in R[X]$ is reducible then it is the intersection of two proper ideals $I_1, I_2$. We can assume these two ideals are principal, so $0 = (f) \cap (g)$. We write $f$ and $g$ in the following form $$f = X^r (a_0 + a_1X + \cdots + a_nX^n), a_0 \neq 0$$ $$g = X^s (b_0 + b_1X + \cdots + b_mX^m), b_0 \neq 0.$$ Here we choose $f$ and $g$ such that $m+n$ is minimal. Replace $f$ and $g$ by $X^sf$ and $X^rg$, respectively.
Fact: Let $f = X^rf'$ and $g = X^sg'$. Then $(f) \cap (g) = 0$ if and only if $(f') \cap (g') = 0$.
proof. The "if part" is clear since $(f) \subseteq (f')$ and $(g) \subseteq (g')$. For the only if part, suppose $(f') \cap (g') \neq 0$. We have $(X^{r+s} f') \cap (X^{r+s}g') = X^{r+s}((f') \cap (g')) \neq 0$. Thus $(f)\cap (g) \neq 0$ since $(X^{r+s}f') \subseteq (f)$ and $(X^{r+s}g') \subseteq (g)$.
Using the above fact we can reduce our assumption to the case $$f = a_0 + a_1X + \cdots + a_nX^n, a_0 \neq 0$$ $$g = b_0 + b_1X + \cdots + b_mX^m, b_0 \neq 0.$$ Now, if $m+n = 0$ then $f, g \in R$ this contradicts with our assumption that $(0)$ is irreducible. So $m+n> 0 $ we assume that $m \ge n$ and so $m>0$. Choose $0 \neq c \in (a_0) \cap (b_0)$ in $R$. So $c = da_0 = eb_0$ for some $d, e \in R$. Replace $f$ and $g$ by $df$ and $eg$, respectively, we can assume henceforth that $a_0 = b_0$.
By the minimality of $m+n$ we have the following.
Claim 1: Let $r$ be an element of $R$ such that $ra_0 = 0$ (and hence $rb_0 = 0$). Then $rf = 0$ and $rg = 0$.
proof. If $ra_0 = 0$ and $rf \neq 0$, then we replace $f$ by $rf$. Since $(f) \cap (g) = 0$ we have $(rf) \cap (g) = 0$. However $$rf = 0 + (ra_1)X + \cdots + (ra_n)X^n \neq 0.$$ Thus $rf = X^tf'$ with $\mathrm{deg}(f')<n$. By the Fact we have $(f') \cap (g) = 0$. This contradicts with the minimaityl of $m+n$.
Using Claim 1 and the induction one can prove the following.
Claim 2: Let $h = c_0 + c_1X + \cdots + c_k X^k$ be a polynomial satisfying that $hf = 0$ (resp. $hg = 0$). Then for all $i = 0, ..., k$ we have $c_if = 0$ (resp. $c_ig = 0$).
By Claims 1 and 2 we have.
Claim 3: $hf = 0$ if and only if $hg = 0$.
We continue our proof. Let $g' = g-f$. Since $a_0 = b_0$, we have $g'$ can be written as $$g' = X^{m -m'} (b_0' + b_1'X + \cdots b_{m'}X^{m'}), b_0' \neq 0, m' < m.$$ By the minimality of $m+n$ and the Fact we have $(f) \cap (g') \neq 0$. Thus there are polynomials $u,v,w$ such that $$0 \neq w = uf = v(g-f).$$ If $vg \neq 0$ then $vg = (u+v)f \in (f) \cap (g)$, a contradiction. Therefore $vg = 0$. By Claim 3 we have $vf = 0$ so $w = 0$. This is also a contradiction. The proof is complete.