'Fixed cost of input is ignored in the study of the law of increasing return'. Do you agree?

No, it is not correct. In fact, fixed cost plays an important role in deriving increasing returns from variable inputs. A firm can leverage its fixed cost to derive better returns due to improved productivity of resources, e.g. rent is paid every month for the factory space which is fixed cost.
By employing more labours, the production can be maximised and returns of paying rent as fixed cost can be increased. Hence, fixed cost is not ignored in the analysis of law of increasing returns to variable factor.