Question. Does there exist a GCD domain $R$ in which the equation $aR \cap b R \subseteq \mathrm{lcm}(a,b) R$ is untrue?
Remark. The superset direction is always true: since $a \mid \mathrm{lcm}(a,b)$, thus $a R \supseteq \mathrm{lcm}(a,b)R.$ Similarly since $b \mid \mathrm{lcm}(a,b)$, we deduce $bR \supseteq \mathrm{lcm}(a,b)R$. Combining these results, we infer that $aR \cap b R \supseteq \mathrm{lcm}(a,b) R.$
The least common multiple of $a$ and $b$ in a GCD domain is an element $\def\lcm{\operatorname{lcm}}\lcm(a,b)$ with the property that whenever $a\mid x$ and $b\mid x$, we get $\lcm(a,b)\mid x$. In terms of ideals, this says that if $x\in aR$ and $x\in bR$, then $x\in\lcm(a,b)R$; in other words, $aR\cap bR\subseteq\lcm(a,b)R$.
Combining this with your observation that $a\mid\lcm(a,b)$ and $b\mid\lcm(a,b)$, it follows that $$ aR\cap bR = \lcm(a,b)R $$ for all $a,b\in R$ when $R$ is a GCD domain.
