The image below is from the wiki winglet page shows the wingtip vortices with and without wiglets. It is producing lift to oppose wingtip vortices and as a result adds a horizonal force component (Think Newton equal and opposite).
Obviously the results of this force are highly geometry dependent, as winglet could be up, down, or both. In the case of the photo, the upward winglet is pushing air outward and consequently has a resultant force oriented towards the aircraft. This creates a load and a moment that must be structurally supported by the wing and root of the wing; just as if it were a cantilever light pole opposing gravity. This moment is in addition to the much greater lift forces on the wing that are more or less operating in the same direction. Basically there is more load, so you need to build the wing stronger to accommodate it. Putting it into perspective though, a properly designed winglet will have a very small force (say less than 5%) relative to the lift of the wing.
The wing will not stiffen because of the winglet or this force. Excluding the geometric complexities of a tapering wing, it will somewhat follow the cantilever beam deflection equation where the deflection is proportional to load.