I happened to do just that.
I took an old hand pump and an inner tube to the bath tub. Submerged the pump and pre-filled the tube with water, complete with burping out any air bubbles. With the tube moderately full and free from air, I put it on the rim with the tire. Then the final pressure up.
The first thing to notice is that the pump operates with lots of resistance (water flows a lot slower than air through those tiny parts).
As I pushed each stroke of the pump, the resistance difficult yet constant. I kind of expected the resistance to increase as I pressured up the tire, but it didn't. It was a solid stopping point, halfway through the last stroke the tire wouldn't take any more water. (I shouldn't have been surprised, water is one of the least compressible substances around!)
Now out for a ride to test this out!
Indeed to carry the bike the front was a bit heavier (heavier is exactly what most cyclists would not want; but hey, lets see if there's some benefits).
Oddly, the bike didn't feel all that different. Just slightly front heavy. But once I got some speed the steering became really quite stable. A ride through the forest and I could quickly pin point my leans for corners, it was soooo smooth! Like the steering of a motorbike; I assume the extra rotational inertia in the front wheel added to the smoothing gyroscopic effect.
Mucking about, I tried a wheelie. It took a bit of getting used to for lifting it up, but once up it was also very very smooth. I was never that good an wheelies, but now I seemed to be able to ride for as long as I wanted. Try balancing a baseball bat with the heavy side down, it's actually much easier with more weight at the top! And the extra gyroscopic effect made it easier to steer the wheelie.
When stopped at a light or something, pick up the front tire and it would slowly start to rotate on it's own.
It was fun, so I kept it.
The next phenomenon was a puncture. sure enough, one day riding on a nice sunny day... I feel tiny sprinkles of water; I had a tiny tiny leak and I was actually spraying a very fine stream. The tire hadn't lost much though, so I double timed it to my destination in hopes of avoiding changing in the field. On arrival, the tire seemed really quite full still. When off the bike the tire really wasn't getting flat. I left it. I continued to ride for a week or so, sometimes the leak would come back... sometimes it would go away.
I'm speculating, as I only did this once... but it occurred to me that with water being incompressible, the small amount of leaking water was very little volume but nearly all of the pressure. The tire was still full and held it's shape, but no pressure to force the leak when I wasn't on it (it was obviously a very small leak).
Eventually I had to add some air, now and again (too lazy to put water in, and too curious to see how bad the water/air mix would be). Soon the water/air mix was sloshing inside and causing a lot of resistance, and that became the end of my experiment.
In the end I did not see any improvement for hill climbs, or maintaining pedal momentum; only wheelies and cornering.