I have Dura Ace 7801 10-speed wheels. Can I install a Shimano 11-speed free hub onto the 7801 hub?
No for a whole lot of reasons, the most decisive of which is that the 7801 hubs are among the only (along with WH-6600) ones Shimano's ever made that use a drive ring in the hub shell to engage the pawls, as opposed to the pawls and the ring they engage being all internal to the freehub body. In other words the design is completely different from any of the 11-speed freehub bodies.
The hub design from WH-7801:
An 11-speed FH-9000 Dura Ace, for comparison:
If this is one of the 20-spoke carbon 7801 wheels that take standard nipples, you could just swap in a new hub.
As for the larger discussion of whether this is possible with Shimano hubs:
Among the Shimano hubs that use a standard M10x1 axle, until 11-speed came along there was a frequently re-used, semi-standard 10-spline attachment scheme for the freehub body, so in many cases there is some freehub body interchangeability, at least in a pinch as there are some other considerations involved to make it work well. (The nonstandard/oversize axle hubs, which are numerous at this point, generally all need the exact replacement freehub body, although there may be some exceptions or hack possibilities now or in the future.) However, the 11-speed hubs using standard axles that have so far been released, FH-5800 and FH-CX75, use a version of the same design that has 15 splines on the hub shell instead, meaning no interchangeability with a shell with only 10 splines.
FH-5700, 2nd generation of 10-speed 105. Freehub body attachment representative of many other Shimano hubs:
FH-5800, 1st generation of 11-speed 105:
You absolutely can put a 11S cassette on a 10S hub but you will need a little help. Dura ace, Ultegra, 105, Force and rival cassettes can be modified to perfectly fit a 10S hub. The Wheel stays intact, no modification needed on hub re-dish of the rim.
The back of the cassette needs to be machined by a qualified machinist. All 11 speeds are 100% functional like they should.
I have been doing this for a year now and I am getting 100% satisfaction from my clients.