I think Aabaakawad's link gives a complete answer, but to give an astronomy for dummies answer, there's nothing about a white dwarf that causes a planet's orbit to decay at least, not directly. Your article (I've pulled quote below the caption):
Slowly the object will disintegrate, leaving a dusting of metals on
the surface of the star.
That's only talking about this particular situation and there's a difference between disintegrate and decay. This planetoid is enormously close to the white dwarf. So close, that what we think of as normal white dwarf/planet dynamics (very cold) is no longer true. This planetoid is slowly being vaporized.
Looking at the orbital period of 4.5 hours (about 1948 orbital periods in 365.25 days). The orbital distance to orbital ratio relation is exponential to the power of 2/3 (this varies a bit due to eccentricity, but it's generally correct), so an orbital period 1948 times faster means about 156 times closer, and giving the white dwarf equal mass to our sun, that puts the planetoid at a bit under 1 million KM. If this white dwarf is lighter than the sun, the planetoid would need to be even closer. That's close to the Roche limit and would be inside it if the planetoid wasn't dense and rocky/metallic.
If we estimate the white dwarf to be about the size of the Earth, which is a common size given for white dwarfs, An Earth sized object from 1 million KM would be larger in the sky than the sun appears from Earth, and presumably quite a bit hotter than the surface of our sun too, so this isn't a tiny white dwarf in the sky from the perspective of the planetoid. It's a blazing furnace of a sun, so hot, it vaporizes metallic gas and dust off the surface of the planet.
The article mentions this (end of page 3), that Poynting-Robertson drag see here and here, and that may be a factor in any orbital decay in this scenario. The article is clear that there's a good deal of uncertainty on with that effect, and that only affects tiny particles, but enough tiny particles could create a drag over time. . . . (maybe). The general scenario with this orbit is a planet scorched and as a result, is losing material. It's likely the very high heat that's driving any orbital decay, not gravity.
Gravitational decay / orbital decay does happen, usually much more slowly. That's probably not what's happening here.
There are some interesting orbital effects that can happen when a main sequence star goes red dwarf and later when it creates a planetary nebula, significant increases in tidal forces due to the star's greater size in the first case and increased drag in the 2nd, but at the white dwarf stage, there's no significant orbital decay effects.
Update:
Why not Poynting Robertson drag and Orbital decay effect the planetoid
when white dwarf was a star or even red giant? Is there any
"interesting orbital effects" when a star undergoes red giant?.Can you
update your answer to summarize the forces and their effect on the
planetoid in each phase of the star.
and also what do you mean by orbital decay? Does it have something to deal with the Roche limit.
OK, I think, having read more about it, Poynting-Robertson effect only matters when the orbiting objects are very small. I've linked it twice above, but the simple explanation is that objects in orbit move and so any light or debris from the sun hits the moving object at an angle, not direct on. If the object is small enough, this over time drives the dust and maybe grain of sand sized particles into the sun. This doesn't affect larger objects, so it's not really relevant to any planets or planetoids.
As far as "interesting red dwarf" effects. That really has to do with tides. Using the Moon/Earth example, the Moon creates tides on the Earth, a tidal bulge towards the Moon, but because the Earth Rotates faster than the Moon orbits, this tidal bulge is always ahead of the Moon and this creates a gravitational tug on the moon that pulls it away from the Earth - very slowly.
The same thing happens with planets around stars, but even more slowly, lets pretend it's just the Earth and the Sun - a 2 body system (in reality, with several planets it's much more complicated), but just Earth and sun, teh Earth creates a tidal bulge on the sun, the sun rotates ahead of the earth, this causes the earth to very slowly spiral away from the sun - so slowly that it might take a trillion years for the Earth to spiral away.
Now when the sun goes Red Giant, the sun is essentially the same mass but much more spread out and parts of it, much closer to the Earth and less gravitationally bound to the sun. This creates a far larger tidal tug. Also, as the sun expands it's orbital velocity drops, because orbital momentum is concerved, so when the Sun is Red Giant, the tidal bulge will be behind the Earth which drags it in towards the Sun. Due to the size and proximity of the Red Giant star, this draws the remaining near-by planets towards the sun fairly quickly, at least compared to main sequence stages which, provided the sun rotates faster than the planets orbit, has a much smaller outward tidal pressure on the planets.
And when the sun goes planetary nebula, any debris in the planet's path can also cause the planets to slow down slightly - the precise process there I'm less clear on, but in general, any orbital debris creates drag and can slow down a planet's orbit. This may be a key factor in the formation of hot jupiters, cause they can't form close to their suns but enough orbital debris can drive them in closer to their suns. (or planet to planet gravitational interactions can too).
That's the gist of the Sun-Planet orbital relation. When the sun is young, planets are mostly driven outwards, and young suns can have far greater solar flares and stronger solar wind. How much that effects the planets, I'm not sure.
During the Main sequence stage, stars tend to push planets outwards (unless they rotate very slowly, in which case the tidal effect is reversed), but this tidal effect is very small and very gradual.
During the Red Giant stage, stars tend to drag planets in wards, and I assume, during the planetary nebula stage as well. This effect is larger for closer planets.
You also asked about Orbital Decay - if you click on the link, there are examples of that. That probably gives a better explanation than I could. In general, Orbital decay happens very slowly unless you're talking Neutron Star or Black hole in which case the relativistic effects can cause orbital decay to happen quite fast. There's nothing about a white dwarf star that would cause faster than normal orbital decay but a white dwarf star would lose any tidal bulge tugging that a main sequence star has, so there would be essentially no tidal outwards pressure either which could in theory speed up decay cause you've lost a small outwards pressure but you would still have any debris or space dust clouds causing a small inwards pressure. (if that makes sense?)
That's my layman's explanation anyway.
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