May one tear open and use bandages and sanitary napkins on Shabbos?
Firstly, one needs to ensure that there is no tearing on letters or designs when opening any package.
If it is possible to open the package without ripping the letters then one may attempt to do so even if there is a chance that he may accidentally tear the letters.
The concerns of opening a band-aid or diapers for the first time on Shabbos touch on either the Melochos of מכה בפטיש or קורע or תופר. There is a Machlokes whether these may be stuck or unstuck on Shabbos. Therefore, one should open them before Shabbos, unstick them, and then they may be resealed. Afterwards they may be unstuck and stuck on Shabbos as needed.
See Shulchan Aruch Admor Hazaken Orach Chayim 317:6 & 340:13. Ibid. 17.See also Minchas Yitzchak 5:39 (who rules that it is forbidden to open the plastic of a band-aid).
The solution of opening the diaper once and then closing is offered by Shemiras Shabbos Kehilchasa 15:87.
Lenient opinions, see Beir Moshe 1:36 and others.
See also references here.