Question: I am invited to a Shabbos Sheva Brachos in an area where there is an Eruv. Can I bring a gift on Shabbos?
Answer: When one gives a present, the item transfers ownership which is akin to a transaction. Therefore, one is generally forbidden to give others presents on Shabbos (Mishna Berura 306:33).
One may give a present that may be used on Shabbos, such as food. One who receives a bottle of wine, for example, is not obligated to open it that day. The Shulchan Aruch (OC 323:7; Mishna Berura 323:34) allows one to give a new dish that requires tevila to a gentile on Shabbos and subsequently borrow it off him, thus permitting the dish for use.
Likewise, the Beis Yosef (OC 527) allows one to give a gift on Shabbos if it will be used for mitzva purposes. This is the basis for allowing one to give their lulav and esrog to another as a gift on the first day (or 2) of Sukkos.
The Magen Avraham (OC 306:15) question the practice of giving presents to a chassan who delivers a speech on Shabbos. The Chasam Sofer (Hagaos Hachasam Sofer OC 306), however disagrees as such a gift serves a mitzvapurpose by giving true kavod to the Torah. The Aruch Hashulchan extends this, allowing one to give wedding presents at the Shabbos Sheva Brachos, as doing so brings simcha to the chassan and kalla; itself a great mitzva.
Likewise, R’ Ovadia Yosef (Yechave Daas 3:21) allows giving Bar Mitzva presents on Shabbos (especially Seforim that may be used that day) as they can serve to encourage the boy in his Torah learning and religious lifestyle.Nonetheless, when giving (non-food) gifts, R' Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 29:29) advises having a third party acquire the gift on behalf of the one receiving the gift. Alternately, they should intend not to acquire it until after Shabbos.