Apparatus: CL 19th; animate objects & continual flame
dispel magic + PaO
So you dispel magic targeting the apparatus (DC 11+19= 30), lasting for 1d4 rounds.
According to RAW, the apparatus is now (momentarily) non-magical, which at this point is nothing more than an oversized 500lbs. trashcan with a hidden lid (Search DC 20). PaO can now be successfully cast on the apparatus. The trashcan is changed in a non-magical, 90.000gp acorn (fine object) for 1 hour (+2 factor for Int).
At a certain point in time, the dispel magic -effect ends. PHB-172:
One Effect Makes Another Irrelevant: Sometimes, one spell can render a later spell irrelevant. For example, if a wizard is using a shapechange spell to take the shape of an eagle, a polymorph spell could change her into a goldfish. The shapechange spell is not negated, however, and since the polymorph spell has no effect on the recipient’s special abilities, the wizard could use the shapechange effect to take any form the spell allows whenever she desires. If a creature using a shapechange effect becomes petrified by a flesh to stone spell, however, it turns into a mindless, inert statue, and the shapechange effect cannot help it escape.
Now it becomes interesting; why not another spell AND what is the later spell exactly? The dispel magic -effect ending, thus the activation (casting) of animate objects & continual flame OR the PaO?
I perceive PaO as the later spell, mainly because dispel magic has a duration (1d4 rounds in this case). If this is actually the case, the result would be that the PaO -effect is dispelled (for the preq's are no longer valid), returning the acorn into a (fully functioning) Apparatus after 1d4 rounds.
If the ending of dispel magic (thus the casting of animate object + cont. flame) is treated as the later spell(-effect), the acorn would become an animated, 90.000 gp acorn that glows the same way a torch would, for 1 hour. After 1 hour, the acorn would turn into a animated 500 lbs. trashcan that glows like a torch and has pincers. Note: this acorn/Apparatus is (ONLY!) under control of the one casting animate object, which is the creator of the Apparatus. As a DM, I find this option rather amusing.
PaO is the later spell. After the dispel magic-effects expires (1d4 rounds), the acorn would instantly return to a fully functioning Apparatus. I think this is RAW. If the acorn is actually carried (within a pocket or an already well-fill BoH for example), this option can have rather unexpected results as well.
It mostly depends on how one interprets that specific sentence about the "recovery" of dispelled magic items, IMHO.
However, because -AFAIK- there's no related errata on the subject for PaO or dispel magic or the PHB-magic section.., I'm sticking to the basics (inherent magic item dispels later PoA-effect) here.