Juliano, I am not sure what you are referring to with "River grape vine" however, generally vines that support themselves on other structures via tendrils (such as the Clematis and common grape) will not harm their host unless they grow so aggressively as to smoother the entire canopy, interferring with photosynthesis.
If this grape is the type of vine that winds around the stems and trunks then this can be an issue as it can cause girdling and constriction.
So I would say as a general rule, vines with tendrils, holdfasts and aerial roots are not going to be too much of a problem, unless as stated above they simply overpower the tree. If you wish to remove these vines, manually or with hand pruners is best. Resist the urge to whack away with an axe or chainsaw to avoid injuring the tree itself.
Sylvia