Suppose the objective implicitly imposes non-negativity constraint, say, the objective is sum of square roots of the decision variables. Is it necessary to consider the inequality constraints imposing nonnegativity and have dual variable associated with them? Then those dual variables and those inequalities will appear in the lagrangian and the dual function.
It depends. What are you trying to accomplish? (By the way, I hope you are maximizing that sum of square roots. Otherwise your model is not convex.)
If you just want to compute the Lagrange dual, then you do not have to add explicit non-negativity constraints to the primal problem. Of course, as is often the case when computing the dual, you may end up with implicit constraints in the dual function. You will have to decide whether it is natural to move those implicit constraints out of the dual function and make them explicit dual constraints or not.
If you are trying to solve the problem, using some method that requires the dual, you may need to add the constraints. That is because, for the case of square root at least, the gradient of the objective will be infinite at the origin. Some algorithms won't be able to deal with that very well. For instance, if you employ a barrier method, I'd recommend going ahead and adding explicit non-negativity constraints.