Generally speaking, Halacha forbids attaching two materials together with glue on Shabbat, as this constitutes a "Tolada" (derivation) of the Melacha (category of forbidden activity) of "Tofer," sewing. Sewing is forbidden on Shabbat due to the sewing that was necessary in making the Parochet (curtain) in the Mishkan, and this prohibition includes all instances of attaching two materials with a third substance. Seemingly, then, closing a diaper with adhesive straps would be forbidden due to the prohibition of "Tofer."
However, several authorities, including Chacham Bentzion Abba Shaul (Jerusalem, 1924-1998) and Chacham Ovadia Yosef, permit closing the adhesive strips on diapers because the strips are attached only temporarily. Since the individual closes the strips with the intention of opening them again to remove the diaper several hours later, closing the strips does not fall under the prohibition of gluing materials on Shabbat, which applies only to attaching materials permanently.
It is forbidden, however, to tape a diaper closed after removing it from the child on Shabbat. Since nobody intends to open the diaper again after it is removed, taping it closed at that point constitutes a permanent gluing, would is forbidden on Shabbat.
Chacham Bentzion Abba Shaul pointed to yet an additional problem concerning the use of diapers on Shabbat. When a person opens the adhesive strips to use the diaper on Shabbat, he at that point renders the diaper suitable for use, which would violate the prohibition of "Makeh Patish" making an article or utensil usable on Shabbat. Therefore, Chacham Bentzion advises one to prepare diapers before Shabbat by opening the adhesive strips, thereby rendering the diaper usable already before Shabbat so that it may be used on Shabbat.
Summary: One should open the adhesive strips on diapers before Shabbat, and he may then use the diapers on Shabbat. It is forbidden, however, to close the adhesive strips after removing a diaper from a child on Shabbat.
