Would dropping a white dwarf or a "still-hot" stellar core into a large planet or brown dwarf create a star?
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.