I would like to find out what the gravitational effects would be on:
A self-expanding, tubular ring built around the equator.
As the ring expands, it should levitate away from the surface of the planet and into space.
Is there anyone who has any interest in this idea?

What would it be made of? Hopes and dreams?

Sounds interesting...What would it be made of and will it be interplanetary and orbit above the atmoshpere or a stationary object that runs along earth? Because you will need some kind of Anti Gravity force to keep it in the conditions you want.

Let me take a guess at what you mean. You build a pipe laying flat on the ground all the way around the Earth (forgetting things like mountains for the moment). Each section can concertina itself into a longer section. Now because circumference=diameter*pi, if one section concertinas out another 3 feet then the circumference increases by 3 feet which means its diameter is now roughly 1 foot greater than the Earth so will it lift itself the extra 6" clearance all the way around the Earth and appear to be levitating? To take it further, if all sections start expanding themselves and adds another 3000 miles to the circumference, will the entire ring raise itself off the ground by roughly 500 miles?

In theory, yes. In practise, no way in the world. First, consider how much something like this would weigh. I'll take a guess and say mega billions of tons. ALL of this is trying to fall down to the Earth at once so it has enormous compression loads at every point. Even close to the surface (say 6") it would be the same thing as a tower roughly 25,000 miles high. It'd need to be built out of "scrith" (read Ringworld by Larry Niven).

It would also be completely unstable in that any disturbance (the moon's gravity for instance) would move it slightly. Any movement means that the closer part to the surface continues to get closer until it crashes into the ground (read Ringworld).

Hopes and Dreams?
Isn't that how all ideas start?

Since the object in question is continuous around the Earth, the gravitational forces pulling on one side would have an equal force acting on the opposite side. If cables were attached at various locations around the globe, a pull an one side should cause lift on the other, theoretically. As well as stabiltiy.
Further, maybe it could be spun once in place to create artificial gravity.

To me, this seems much more feasible than a space elevator climbing a thin ribbon of cable extending hundreds or thousands of miles out into space. What would the maximum load be before you begin pulling down (reeling in) the geostationary satelite at the upper end?

well there's other things to keep in mind, the mountains that were mentioned and then ignored for one, plus gravity isn't the same all over the earth.

The mountains are probably less of an issue than the oceans.

Maybe it could be constructed a few degrees off the equator to avoid most of the mountains. Paths could be cut where needed.

Again, by attaching cables at strategic locations, the uneven forces of gravity could be compensated for. As well as compensating for the weight of the loads that would be attached for lifting into space.

This tube could be made with carbon nano-tube material to minimize weight and maximize strength.

I'll agree, in theory this could be done. But, to secure the funding, manpower, international cooperation, resources, and other things would be nearly impossible.

But, it's a fun thought to entertain, I'll give you that.

I have to say that I think downunder's right about the compression loads at every point. To give an analogy, say you have a bagel. Let that bagel represent the ring. Since the Earth's gravity is pulling each section of the ring down toward the center of the Earth, this can be analogized by wrapping your two hands around the bagel's circumference and pushing the bagel towards its center. Eventually, the bagel will collapse and you'll have a ball of mush.

By the same token, the ring won't be able to stand the enormous compression forces caused by gravity's pull and will buckle and slam back down to Earth even if you expand it by only a relatively short distance. You could put supports under the ring to push it outward, but the sheer number of supports you'd need to have would probably make the ring pointless anyway.

Of course, if you did have scrith, then it'd be okay then.