If an Inner Sphere Sun went Supernova would there be any warning?
Depends on the progenitor:
1. A massive white dwarf that accretes mass from a companion. Runaway fusion will set in after the dwarf reaches about 1.38 solar masses. If this progenitor would have been visited, its threat would certainly have been noticed. Even a rough prediction of the expected time before detonation could have been made even centuries in advance. On the other hand from afar, there would hardly have been a warning because white dwarfs are faint and numerous. Nobody would have noticed.
2. A binary star merger, likely also involving white dwarfs. As before, if the system had been examined, orbital calculations would have revealed a threat, including rough predictions of remaining lifetime. Likely also centuries in advance. From afar, the rather rapidly increasing radial velocity might have been detected, as long as the binary was bright/close enough. Though it might as well be overlooked, considering the large number of multiple system (+ planets) that would have to be observed.
3. A massive star facing a core collapse event. These things are bright and rare. Shortly before (no idea how long but possibly decades) the core collapse, the star might show some strong pulsations because the initial fusion process does not stop instantly but step by step. This would be noticeable even far a few hundred parsec away (Note that this means, the star might already have detonated while it has not seen it yet).
Would anyone outside the system know about it? (when it happened any long range effects? I know light would still take time to pass the distance between the stars but would it be noticeable on sensors or cause problems for navigation - theory?)
Per current knowledge of physics, no. Neutrinos take years to reach the next systems where they could be detected. Light arrives a couple of hours later.* But there is no instant way of getting the info to other star system.
What Kearny-Fuchida physics say about this is up to you. One might construct a in-game theory that a nearly instantly noticeable KF-space shock is detectable by all near jumpships.
*No not because light is slower than neutrino but because the neutrinos leave the detonation mostly unhindered at almost lightspeed while the light takes several hours to reach the surface of the former star because of the strong scattering in the dense matter.
What would be left?
The original star is gone, not even a thin cloud remaining. An envelope is expanding at high speed. At first it is of course extremely hot dense. Therefore, any potential nearby planets are evaporated. Further out, the cores of former gas planets may survive in some twisted form. Within a couple of AU, the neutrino flux will instantly saturate matter with radioactive species. Well, the range of this depends on the exact scenario of the SN. Type Ia supernovae, the 1st scenario above, are presumed to produce enough hard radiation to mess with the ionospheres and destroy ozone layers of planets within a radius of 1000pc. So basically, the whole Inner Sphere could be wiped out unless people notice the supernova and begin evacuation. Not impossible, considering that there are decades or even centuries for the evacuation before the radiation front arrives. Some Type 2 supernovae might be much less deadly and only affect a sphere of 10pc.
Could a JumpShip still enter the area?
Jumpships can jump everywhere, as long as the gravity field is not too complicated to be considered in the jump sequence. Therefore, as long as one does not jump into or close to the expanding gas shell you are okay. Well, unless you wish to define a Kearny-Fuchida-physics effect that prevents a jump into the system.