Let $M$ be a closed (smooth) Riemannian manifold. Then we have the following existence and uniqueness theorem for the heat equation on $M$, which is considered more or less a standard result:
Let $0<\alpha<1$. $f\in C^{\frac{\alpha}{2},\alpha}([0,T]\times M)$ and $u_0\in C^{2+\alpha}(M)$. Then there exists exactly one $u\in C^{1+\frac\alpha2,2+\alpha}([0,T]\times M)$ s.t.
$\partial_t u-\Delta u=f$ on $[0,T]\times M$
$u(0,.)=u_0$ on $M$.
Here, all maps have values in $\mathbb{R}$.
Is the above theorem still true, if all the maps have values in $N$, where $N$ is another closed Riemannian manifold and the Laplacian $\Delta$ is replaced by the tension $\tau$? If yes, can someone give me a reference for that kind of result?
The obvious generalization doesn't really make much sense when $f$ is non-zero: $f$ should be a section of $u^* TN$, but you don't know $u$ yet. It's possible to replace it with $f \circ u$ for $f$ a vector field on $N$, but since this is of a significantly different character to $f$ in the original problem and there are already issues for $f=0$, I will just discuss that case.
The PDE $\partial_t u = \tau(u)$ is Harmonic Map Heat Flow, which is extremely well-studied. Existence and uniqueness for some time interval $[0,\epsilon)$ works, but you need some more assumptions to rule out singularities. The famous Eells-Sampson theorem (original paper) tells us that when $N$ has non-positive curvature, the solution exists for all time and converges to a harmonic map. Without such a curvature assumption, $u$ can blow up in finite time, even in the $2 \to 2$ dimensional case: see e.g. this paper.