Besse (1978, p.202) has the spectra of compact rank 1 symmetric spaces (CROSSes). In addition to $\mathrm S^n$ due apparently to Heine (1863, §19; 1878, §128), this gives $\mathbf{RP}^n$, $\mathbf{CP}^n$, $\mathbf{HP}^n$ and $\mathbf{OP}^2$.

**Edit:** Also, for $M$ a compact semisimple Lie group it is well known (due apparently to Freudenthal (1954))^{1} that the Laplacian (= Casimir) acts on the $\lambda$-subspace in the Peter-Weyl decomposition $L^2(M)=\smash{\bigoplus_\lambda V_\lambda^{\phantom*}\!\otimes V_\lambda^*}$ by the scalar $c(\lambda):=\smash{\|\lambda + \rho\|^2-\|\rho\|^2}$; so these are its eigenvalues. $(\lambda$: dominant weight; $\rho=\frac12\!\sum\limits_{\alpha > 0}\alpha$; $\|\cdot\|$: Killing norm.)

**Further edit:** The literature contains quite a few more cases than the answers so far. As no single source or search word easily returns them, I list here what I found (others’ answers not repeated):

First, the Casimir method above extends to give the spectrum of the normal metric on $G/H$ ($G$ compact semisimple, $H$ closed). In fact, by Frobenius reciprocity, $V_\lambda$ occurs in $L^2(G/H) = \operatorname{Ind}_H^G1$ with multiplicity equal to the dimension of $V_\lambda{}^H=\{H$-fixed vectors in $V_\lambda\}$. So the eigenvalues are exactly all $c(\lambda)$ for $\lambda$ such that $V_\lambda{}^H\ne0$. After spheres, this method was applied to:

Stiefel manifolds $\mathrm{SO}_n\,/\,\mathrm{SO}_{n-m}$ by Levine (1969, p.519), Gelbart (1974), Strichartz (1975).

CROSSes ($\mathbf{RP}^n$, $\mathbf{CP}^n$, $\mathbf{HP}^n$, $\mathbf{OP}^2$) by Berger & al. (1971, pp.159-173), Cahn & Wolf (1976).

Flag manifolds $G\,/\,T$ ($T$: maximal torus) by Yamaguchi (1979, p.110).

Grassmannians $\mathrm{Gr}_2(\mathbf R^n)$ by Strese (1980, p.78) and Tsukamoto (1981).

Aloff-Wallach spaces $\mathrm{SU}_3\,/\,\mathrm S^1$ by Urakawa (1984, p.984) and Joe et al. (2001, p.417).

Symmetric spaces $\mathrm{SU}_n\,/\,\mathrm{SO}_n$ by Gurarie (1992, p.253).

Grassmannians $\mathrm{Gr}_n(\mathbf C^{n+m})$ and $\mathrm{SU}_{n+m}\,/\,\mathrm{SU}_n\times\mathrm{SU}_m$ by Ben Halima (2007, pp.546, 549).

Secondly, some cases yield to other methods:

Lens spaces $\mathrm S^{2n-1}\,/\,\mathbf Z_p$ by Sakai (1976, p.256).

Hopf manifolds $M_\alpha$ by Bedford & Suwa (1976, p.261).

Berger spheres (total spaces of the Hopf fibration $\mathrm S^1\to\mathrm S^{2n+1}\to \mathbf{CP}^n$ with rescaled fiber) by Tanno (1979, p.184).

Jensen spheres (total spaces of the Hopf fibration $\mathrm S^3\to\mathrm S^{4n+3}\to \mathbf{HP}^n$ with rescaled fiber) by Tanno (1980, p.103) and Nilsson & Pope (1983, p.68).

Grassmannians $\mathrm{Gr}_2(\mathbf C^n)$ by Sumitomo & Tandai (1985, p.153).

Riemannian two-step nilmanifolds $G\,/\,\Gamma$ by Pesce (1993).

^{1 }**Note added:** Rogawski–Varadarajan (2012, p. 690) attribute the formula for $c(\lambda)$ to Casimir–van der Waerden (1935; note the reviewer). However, I’m not sure I can find it there...?