Suppose first that $M$ is a compact Riemannian manifold. Let $D^*(M)$ be the $*$-subalgebra of $\mathcal{B}(L^2(M))$ (the bounded operators on $L^2$) consisting of operators $T$ such that for all $f\in C_0(M)$, $fT - Tf$ is a compact operator on $L^2(M)$.
Question 1: Is it true that $D^*(M)=\mathcal{B}(L^2(M))$? If not, what is an element of $\mathcal{B}(L^2(M))\backslash D^*(M)$?
More generally, if $M$ is a non-compact Riemannian manifold, let $C^*(M)$ be the closure in $\mathcal{B}(L^2(M))$ of the set of operators $T$ such that:
1) For all $f\in C_0(M)$, $fT$ and $Tf$ are compact operators on $L^2(M)$;
2) $T$ has finite propagation.
The second condition means that, for $f,g\in C_0(M)$, there is a constant $C$ such that $fTg=0$ whenever the supports of $f$ and $g$ are separated by a distance greater than $C$.
Then $C^*(M)$ contains the finite-rank operators on $L^2(M)$, hence it contains the compact operators $\mathcal{K}(L^2(M))$ (which is the closure of such finite-rank operators).
Question 2: How different are $C^*(M)$ and $\mathcal{K}(L^2(M))$? In particular, what is an example of an operator that is in $C^*(M)\backslash\mathcal{K}(L^2(M))$?
Here is an example for Question 1: Let $a$ an automorphism of $M$ and $Tu:=a^* u$. Then $$[f,T]u(x)=f(x)u(a(x))-f(a(x))u(a(x))= \left((f-a^*f)\cdot Tu\right) (x).$$ Since $T$ is an isomorphism, we have $[f,T]T^{-1}=f-a^*f \in C(M)$. If $[f,T]$ was compact, then also $f-a^*f$ would be compact, which (I guess) only happens, when $f=a^*f$.