I don't believe so, no. I tried to grab something from the house of my non-mayor character on my mayor and it wouldn't let me, but it may be different for online visitors. I don't think so, though.
If you invite people over they can do pretty much anything else, though. Collect fruit, cut down trees, pick flowers, take anything you have sitting on the ground, etc.
No they can't pick up any items from your home. They can however run through your flowers, pick up your flowers, shop the items at your shops on mainstreet as well as retail, they can pick up any items off the ground outside your homes, and shake your trees.
Most people on here kind of have an understanding that when you visit, unless given permission to, you don't run (only on pathways), you do not pick up anything unless instructed to, you also never speak to the villagers and you always ask before doing something such as leaving to go to mainstreet, shopping, or entering homes.