Can I use the {Invisible} tag on an item frame?

As Skylinerw said, item frames cannot be made invisible.

However, you just want to make a floating item. You can summon an item that won't despawn which players can't pick up. On top of that, you can use the NoGravity tag which will allow the item to float in mid-air. Combining these 3 abilities, you can summon the item using this command:

/summon item ~ ~ ~ {Item:{id:stone,Count:1},Age:-32768,PickupDelay:32767,NoGravity:1} 

If you do not like that the item is spinning around, you can use a second method where an invisible armor stand has the item in his head slot using this command:

/summon armor_stand ~ ~ ~ {Invisible:1,NoGravity:1,Marker:1,ArmorItems:[{},{},{},{id:stone,Count:1b}]}