From the results of

Brown, Arlen; Pearcy, Carl, Structure of commutators of operators, Ann. Math. (2) 82, 112-127 (1965). ZBL0131.12302.

an element of $B(H)$ is a commutator $xy-yx$ if and only if it is not of the form $\lambda I + C$ for some scalar $\lambda$ and compact $C$. So one can for instance take $A$ to be the unital commutative algebra generated by an arbitrary compact $C$ to answer your question in the positive.

From the results of

Brown, Arlen; Pearcy, Carl, Structure of commutators of operators, Ann. Math. (2) 82, 112-127 (1965). ZBL0131.12302.

an element of $B(H)$ is a commutator $xy-yx$ if and only if it is not of the form $\lambda I + C$ for some non-zero scalar $\lambda$ and compact $C$. So your algebra would have to contain an element of the form $\lambda I + C$ but not $C$ (unless $C=0$), and so cannot contain the scalars unless it consisted solely of the scalars, which are indeed not commutators by the classical results of Wintner and Wielandt.