We discuss the influence of distributed dislocations on the nonlinear deformation of a hollow circular elastic cylinder. Hydrostatic pressures are set on the inner and outer surfaces of the cylinder. Longitudinal force and torque act on the ends of the tube. The system of equations of the nonlinear continuum theory of dislocations is solved by the semi-inverse method. For the axisymmetric distribution of edge and screw dislocations in the case of an isotropic elastic material, the initial problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the framework of the isotropic incompressible material model, several exact quadrature solutions of the problem of cylinder equilibrium with dislocations under large deformations are founded. These solutions are universal in the class of isotropic elastic bodies. In a number of practically important cases, we have obtained simple explicit formulas. These formulas describe the effect of dislocations on the nonlinear resistance of the cylinder to external loads.