All of the individual Weapon Camos, Completionist Camos, and Weapon Mastery Challenges from Modern Warfare II carry forward into Modern Warfare III.
All Camos: With the exception of the MWII Completionist Camos (Gold, Platinum, Polyatomic, and Orion), every MWII Base Weapon Camo can be applied to any weapon in MWIII, no matter which game the weapon or Camo originated in. That means you can place a MWII base Camo onto a MWIII weapon, and vice versa, assuming weapon and Camo have been unlocked (examples are shown in the previous screenshot).
Modern Warfare II Completionist Camos: These can be placed only onto the MWII weapon you have unlocked the Camo for, as the four Completionist Camos are accessible only within the Customize > Camo Category Menu. MWII Completionist Camos cannot be placed onto MWIII weapons, or vice versa.
Example: MWII player with unlocked MWII weapons: If you’re a Modern Warfare II player with a number of individual Weapon Camos and Completionist camos unlocked, then those will Carry Forward into Modern Warfare III.
Specific Weapon Example: Let’s assume you’ve unlocked some or all of the following M4 AR Camos in MWII (as shown above):
In this example, any and all unlocked content is Carried Forward into Modern Warfare III. Any content you haven’t unlocked becomes available to unlock within MWIII as well, so you can effectively continue your MWII grind into MWIII.
All Camo Challenges carry over from MWII to MWIII, so, for example, you still need to “Get 15 Kills from behind with the M4” to unlock the New Urban Camo (which, once unlocked, can be applied to the M4, or any other weapon from MWII or MWIII).