I would like to ask for references on automorphisms of a modular tensor category, that do not change the objects. Some special cases, such as automorphisms of a quantum double, are also helpful.

For most quantum group categories *all* braided autoequivalences are classified by Cain Edie-Michell in this paper. The kind you're interested in, which is called "gauge auto-equivalences" there, almost never exist. One example where it does happen is Cor. 3.2 for the adjoint subcategory of sl_3 at level 3, but I'm not sure if that one is braided.

Many of the arguments there build on an idea from my paper with Grossman Thm 5.5 which is that if you have a planar algebraic description of your tensor category then gauge autoequivalences in particular give you an automorphism of the planar algebra, and you can often see that no such automorphism exists.

Not much known in general, but in this paper https://arxiv.org/abs/1312.7466, Davydov gives a description of them for Drinfeld Centers of Vec(G).