If a problem has multiple optima, adding small penalties or rewards to variables that do not directly contribute to the true objective can help by breaking ties that the solver would otherwise encounter. Note that I said "can" help, not "will" help. The solver would work through those ties on its own, so the added terms only help if you get lucky in the choice of variables and coefficients. As far as I know, you can also make the solution time greater by an unlucky choice of spurious terms.
I think you may also benefit in some cases by avoid degeneracy in the dual of the LP relaxation. Again, I believe this adheres to the old adage about bear hunting: some days you get the bear, and some days the bear gets you.
One thing, however, is certain: if you make the coefficients of the additional terms too large, you can end up with a solution that is optimal in the modified problem but suboptimal in the original problem.