Answer: The Shulchan Aruch (OC 266:13) writes that if one finds a purse on Shabbos, one mustn’t pick it up.
The Mishna Berura (Biur Halacha 266:13) writes that it is questionable whether the prohibition of muktze is set aside for the mitzva of hashavas aveida. He notes that the Vilna Gaon (OC 586:50) quotes the Mordechai (Sukka 747) who allows one to use a shofar that would otherwise be muktze on Rosh Hashana. Nonetheless, R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 5:42:12) writes that even the Vilna Gaon doesn’t pasken like the Mordechai and the consensus is that one cannot pick up something muktze even to perform such mitzvos.
The Shulchan Aruch (CM 264:1) writes that the halacha is that one may recover one’s own lost objects before bothering with others’ objects. Thus, the Chasam Sofer (OC 82) writes that as one wouldn’t be able to recover one’s own muktze items on Shabbos, one wouldn’t be able to return another’s muktze items either. He compares finding a muktzeh item on Shabbos to finding chametz on Pesach which one must cover rather than move (See Shulchan Aruch OC 446:1; Magen Avraham 446:2).
The Shulchan Aruch Harav (CM Metzia 40) adds that the mitzva of hashavas aveida does not take precedence over certain other mitzvos. Although muktze is assur miderabanan, one still may not pick up a muktze item to perform this mitzva.
In conclusion, one may not pick up a muktze item on Shabbos, even to return someone’s lost item.