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Let $R$ be a Noetherian domain and $M$ and $N$ be two faithful $R$-modules. Is it true that $\operatorname{Ann}_R(M\otimes_R N)=0$?

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No. $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$ are faithful $\mathbb{Z}$-modules, but $\ \mathbb{Q}\otimes _{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}=0$. However this is true if $M$ and $N$ are finitely generated (no noetherian hypothesis needed), see e.g. Bourbaki Commutative Algebra II, §4, Proposition 18.

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