If we built an orbiting platform of sufficient mass (let's say, made from harvested asteroids) that was anchored at both poles of the planet it surrounded, would such an addition to the planet's mass affect its orbit? Further to this, if the sun in this hypothetical system was dying and lowering its temperature and mass over time, could an orbiting platform around a planet be used to intentionally affect its orbit such that the planet remains in the 'habitable zone' of this sun (rather than become colder and colder as the sun cooled down over time) by reducing its orbital radius?
Backup, frameshift time:
You can't anchor your ring to the poles without making it out of handwavium.
Your ring must be at the equator. Anything else will cause torque on it that it can't hope to survive. It must also be basically self-supporting as you simply can't build load-bearing structures at that scale as nothing is strong enough.
Furthermore, it must be anchored to the planet or you will have a catastrophic failure as anything other than a massless ring has negative stability. This means the anchor points must be moving at Earth's rotational velocity.
In the simple case this means in a synchronous orbit. If you trust your engineers enough there is another option: a two-part ring, the outside part rotating with the planet and the inner part rotating well above orbital velocity to provide the force required to hold the outer part up.
As for the end of life of the star--stars get warmer over time. In the "best" case Earth's fate is to become a ball of molten lava, in the worst case it falls in. Only after that does it go out and collapse into a white dwarf. Even if you could push a planet around enough I doubt it's worthwhile to try to live on the feeble light of a white dwarf.
Also, your ring does nothing to alter the planet's orbit.
First of all, the ring you are suggesting would have to be equatorial not polar.
Secondly the sun will expand in the next million years, so prospectively it will get hotter here (and NO: that is not the reason for global warming!)
So now to your actual question: would earths orbit change significantly because of a ring space station?
Reason: Planets - like earth - are following a quite good kelperian orbits, except some very small disturbances because of other planets (mainly Jupiter). So all we would have to consider is earths mass. By collecting extraterrestrial material and putting it into earths orbit, you basically increase earths mass. More earths mass would follow to stronger disturbances. But: earths mass is variable, in fact earth is loosing about 50000 t per year. By adding some mass of the order of some hundred tons, you basically just set earths mass to status it had some years ago. Since earths orbit is quite constant for the last million years, you would have no significant impact on earths orbit.
To expand Loran Pechtel's answer in terms of moving the Earth - having a spinning ring does not change earth's orbit. In overall terms, if no mass leaves the the earth/ring/moon system the center of mass will remain orbiting the sun on the original orbit. You might be able to move that point around with respect to the Earth or moon, but the combined system will not do anything interesting as long as all mass stays bound to the same Barycenter.
If made from asteroids there are a number of ways constructing the ring would change earth's orbit, due various flavors of momentum transfer. The effect would be slight however because Earth is just so big- even if we took Ceres as the largest single asteroid (9.310^20 kg) and 1/3 of the total mass of the asteroid belt and slapped it into Earth (510^24kg) going retrograde and do some basic newtonian momentum math we get an answer that the change in Earths orbital velocity of ~29 kilometers a second is a mere 10 meters per second - where the existing seasonal range is already 1000 meters per second.
Other methods of getting asteroid mass into LEO would be less 'efficient' in changing earth orbital velocity so will have an even lower effect, presumably measurable to suitable instruments (that 10ms will change year length by about an hour) but not useful for chasing the suns habitable zone out and in. That would require moving around an object decent percentages of an earth mass, even if using something like a gravity tractor.
Note also that building a massive orbital ring tethered to earth is a high risk enterprise that has a number of extinction potential events should it fail.