Suppose we have 3 symmetric, positive definite $n \times n$ matrices $A, B, C$. It is known that $A \geq C$ in the sense that the matrix $A - C$ is a positive semi-definite matrix.
Let $D = AB$ and $E = CB$. Is it true that the largest eigenvalue of $D$ must be larger than the largest eigenvalue of $E$?
Yes. This is true as long as $A\succeq C$ and $B\succeq0$. The matrices $A$ and $C$ don't need to be positive definite.
Since $D$ and $E$ are similar to $B^{1/2}AB^{1/2}$ and $B^{1/2}CB^{1/2}$ respectively, and $B^{1/2}AB^{1/2}-B^{1/2}CB^{1/2}= B^{1/2}(A-C)B^{1/2}\succeq0$, we have $$ \lambda_\max(D)=\lambda_\max(B^{1/2}AB^{1/2})\ge\lambda_\max(B^{1/2}CB^{1/2})=\lambda_\max(E). $$