Solar core is 34% of the Sun mass, so sun will continue to be a star, at some point after the event. It will implode and probably intensify the processes compared to previous conditions and to a comparable star with 66% of the Sun mass. Dynamics of collapsing processes, may lead to ejection of plasma, so it definitely not recommended for planets like our at current state of our technological development.
The core of the Sun is considered to extend from the center to about 0.2 to 0.25 of solar radius
$$U = -G\int_0^R {\frac{(4\pi r^2\rho)(\tfrac{4}{3}\pi r^{3}\rho)}{r}} dr = -G{\frac{16}{3}}\pi^2 \rho^2 \int_0^R {r^4} dr = -G{\frac{16}{15}}{\pi}^2{\rho}^2 R^5$$
From Gravitational binding energy wiki article, but in this case core is out, so is out 0.25R
$$U = -G\int_{0.25R}^R {\frac{(4\pi r^2\rho)(\tfrac{4}{3}\pi r^{3}\rho)}{r}} dr = -G{\frac{16}{3}}\pi^2 \rho^2 \int_{0.25R}^R {r^4} dr = -G{\frac{16}{3}}\pi^2 \rho^2 \cdot \frac{1}{5}(R^5-0.0009765625 \cdot R^5)$$
They assume even distribution for mass, which is definitely not the case, but we will continue assumption farther, let assume that a thing called radius will not change significantly for star with 66% of its original mass (not true but lets hope it is enough true for our purposes)
With these assumptions energy stored as kinetic energy/heat energy/etc as result of this collapse will may be like:
$$ 0.0009765625 \cdot \frac{0.4356 \cdot 3GM^2}{5R} = \text{9.68176538754e+37 J}$$
That is a Lot, even compared to 3.828e+26 J/s which sun produces, according Sun Fact Sheet
- I'm not sure in any assumption and calculations here, I'm just try to estimate orders of magnitude of orders of magnitude. Looks like it have potential for nasty things.
The question is -- will it be enough? What happens to supernovas is not because they collapse, collapse itself is consequence of what actually happening, and happening changing the fuel they burn. (I do not posses deep knowledge about the processes there, but this is one of them)
Hydrogen burning is a slow process, if we compare it to other types of thermonuclear reactions, as it can be seen in example with thermonuclear bombs, they do not need such extreme conditions as sun have constantly in the core, and they produce more energy per given mass, then sun does per same mass in same time.
Removing core, have potential to slow down burning, as it contains heavy atoms like maybe carbon which maybe helps to catalyze hydrogen burning.
But this potential energy of collapse will heat hydrogen to higher temperatures and maybe compress it in to more dense state for some amount of time, which may not linearly improve speed of hydrogen burning. Which will lead to expansion of matter, slowing down the reaction and create circumstances to collapse again.
I will not wonder if collapsing/expanding cycles will continue for next million of years. How long it will continue will be question of how good will be that system as oscillator.
During that dance solar ejecta will have place, that is for sure, and it will be spectacular to observe from a safe distance.
Will it act really as supernova - probably not, depends, mmm interesting question. I mean sure not any star will, some stars may really become supernova from that, specially if core removal is done in the way to maximize that probability, but others will not.(basically these who may be supernova in a future they can, who will not, they probably will not)
Will this situation lead to some nasty things happening in the star system in a supernova fashion way of bad and good(depends who and for what uses that). Yes, it probably will have some elements of supernova - energy bursts, plasma bursts etc.
Will it be a apocalypse for star system, for planets probably not, for some one on a planet, probably yes.