You most definitely can and should go to the Duomo to see the mosaics on the floor!!
Let's just be clear about one thing since Donna didn't specify:
----> you can always SEE most of the mosaics in the Duomo at any time of the year, you won't "miss" seeing them!
The "special" opening regards the mosaics under the part that falls directly under the main dome, which are normally covered with special cardboard and pews and chairs are placed over them while the cathedral is used for mass and services. This is the place where people can sit.
During the period that it is ALL UNCOVERED, between August and October, the chairs are all taken out of that area and those mosaics are ALSO uncovered. That's what the special "uncovering" means, that you get to see them ALL. But at other times of the year, you get to see the rest, so definitely do not skip visiting the Duomo in July!