Would dropping a white dwarf or a "still-hot" stellar core into a large planet or brown dwarf create a star? In the Wikipedia article about stellar engineering, I found this quote:

In The Saga of the Seven Suns, by Kevin J. Anderson, humans are able to convert gas giant planets into stars through the use of a &quot;Klikiss Torch&quot;. This device creates a wormhole between two points in space, allowing a neutron star to be dropped into the planet and ignite stellar nuclear fusion.

Now this is definitely impossible, as neutron stars are degenerate and cannot fuse. But what if we dropped a helium white dwarf or extract the hydrogen-burning core of a low-mass star and put it into a large gas giant or non-fusing brown dwarf? Would this trigger fusion or would this fail as well (presumably because the white dwarf is also degenerate)?
The issue is not whether the dense core can fuse, but what fusion processes can occur on its surface. Remember that novas happen when gas accumulate on the surface of white dwarf stars: the high temperature and density is enough to trigger runaway fusion.
Hence this method, where you to magically make a dense core appear inside a gas giant or somehow drop it onto the core, would likely produce a nova effect. Fusion starts, heats up the hydrogen, this speeds up fusion and increases the temperature, and then the whole thing blows up. Likely not enough for a supernova, but not really creating a star as much as just burning up the planet.
The exception may be a low mass star core, where the gravitational gradient and temperature may be mild enough to allow gradual burning. Here unfortunately a lot depends on the assumed magic making the dumping happen, which presumably makes the answer ill defined.