For any $N\ge 2$ and $α=(α_1,\cdots, α_{N+1})\in (0,\infty)^{N+1}$, let $μ^{(N)}_α$ be the Dirichlet distribution with parameter $α$ on the set $Δ^{ (N)}:= \{ x \in [0,1]^N:\ \sum_{1\le i\le N}x_i \le 1 \}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form $${\scr E}_α^{(N)}(f,f):= \sum_{n=1}^N \int_{ Δ^{(N)}} \bigg(1-\sum_{1\le i\le N}x_i\bigg) x_n(\partial_n f)^2(x)\,μ^{(N)}_α(d x)$$ with Domain ${\scr D}({\scr E}_α^{(N)})$ being the closure of $C^1(Δ^{(N)})$. We prove the Nash inequality $$μ_α^{(N)}(f^2)\le C {\scr E}_α^{(N)}(f,f)^{\frac p{p+1} }μ_α^{(N)} (|f|)^{\frac 2 {p+1}},\ \ f\in {\scr D}({\scr E}_α^{(N)}), μ_α^{(N)}(f)=0$$ for some constant $C>0$ and $p= (α_{N+1}-1)^+ +\sum_{i=1}^N 1\lor (2α_i),$ where the constant $p$ is sharp when $\max_{1\le i\le N} α_i \le 1/2$ and $α_{N+1}\ge 1$. This Nash inequality also holds for the corresponding Fleming-Viot process.