The effects of non-conservative forces and nonholonomic constraints on Noether symmetries and conserved quantities of a Lagrange system are studied. When non-conservative forces or nonholonomic constraints are inserted in a Lagrange system, the Noether symmetries and the conserved quantities of the system may vary. Some Noether symmetries disappear, some new Noether symmetries emerge, and under certain conditions, some Noether symmetries will still remain. The conditions under which the Noether symmetries and the conserved quantities of the system will remain are given, and two examples are given to illustrate the application of the results.