The operators of all observables are time-independent in the so-called Schrödinger picture of quantum mechanics. This is not the case in the two other pictures, known as the Heisenberg picture and the Interaction picture. Operators in the latter two pictures are dynamical, their dynamics being governed by the appropriate equations of motion. For details, see the initial sections of Chapter 3 of the book Quantum Theory of Many-Particle Systems, by Fetter and Walecka (Dover Publications, 2003).

The time dependence does not disappear but it appears as another form such as Schrödinger representation, where states become time-dependent and operators are time-Independent. But there are unitary transformations to pass from one representation to another. We can use the evolution operator, for example, to pass from as Schrödinger representation to Heisenberg representation. .

In a way angular momentum (operator) in quantum mechanics is time dependent because it includes Planck constant which has units of Joules.second.

In classical mechanics angular momentum is ⃗L=⃗r x ⃗p. Then Lx=y pz - z py and so on. In quantum mechanics we use operators for px and pz as -i ħ ∂/∂x and -i ħ ∂/∂z and then we get the operator for Lx. Similarly we get operators for Ly and Lz. It is now easy to see why there is no time dependence in these quantum mechanical operators even though classically the expression involves a cross product with linear momentum which involves the "time derivative" dx/dt. That is classically if dx/dt is time dependent so is ⃗L

But remember that in quantum mechanics you have the operator as well as the eigenfunction on which it acts. You are interested in the eigenvalue. The eigenfunction is the wave function which is a product of  a space part and a time dependent part which is a pure phase. We see that L operator does nothing to the time dependent pure phase factor when it acts on the wave function.