Yes, no effect on the sun at all as the others said. This is potentially a way to dispose of them, except that sadly, we have to launch them first. Launching so many missiles at the Sun - they all have to reach not just escape velocity, but also need to counter the huge 30 km / sec orbital velocity of the Earth around the sun.
With existing technology, they'd be doing repeated flybys of Earth Venus, Jupiter and so on before they eventually lose enough delta v to hit the sun.
Just getting them all into orbit though would be hazardous enough. It's not practical sadly.
Instead we have to deal with them here.
And - after the arms reduction talks, then a lot of the previous weapons grade uranium stock piles have been converted to nuclear fuel. About 12% of the nuclear fuel used world wide in recent years comes from Russian nuclear weapons reprocessed into fuel, and exported most of it to the US to be used in US reactors. And the US nuclear weapons also reprocessed in the same way. Our nuclear reactors have been burning tens of thousands of nuclear weapons in this way over the last few decades.
This used highly enriched uranium. There's been research into ways of burning plutonium as well, but not actually being done.
So anyway - that's one idea which seems like it should work is to burn the plutonium and other long lived products to make power until all you have is short lived waste.
Though it's not as easy as it seems. Here is an article goes into the techy details of some of the ways we could burn plutonium and some of the issues with it. Plutonium as an Energy Source
And then turn the short lived waste into synthetic rock, is one of the ideas. With reactors especially tuned so that they burn up long lived waste and generate only short lived waste which would soon be harmless.
Possibly also could use fusion reactors once we have those, to dispose of it. Quite a reasonable section on it in wikipedia here: Radioactive waste
Meanwhile, ironically, NASA has a serious shortage of plutonium fuel for its spaceships. It's the only way you can supply power to a long lived spacecraft like the probe to Pluto, the Pluto Express. Solar power works to Mars, and to Jupiter (just) but hard to use it further afield without huge solar panels.
So, this is just a small amount of plutonium, and well protected. It's not a nuclear reactor, and can't be made into one, because it is Plutonium 238, which is what you have left after you refine the Plutonium 239 for nuclear weapons. And it can be separated out, but is often left mixed up in the high level waste. The plutonium we have come from special programs that separated out the Plutonium 238 for use as a power source for spacecraft, spy satellites etc.
The plutonium is just a source of heat, through radioactivity (without a chain reaction), which is then used to create power. Just so happens to be ideal, just the right amount of radioactivity to keep warm for a long time, but no chance at all of a chain reaction. A rather simple device, none of the complexity of a nuclear reactor, just a heat source and a heat sync, which then can be used to generate a few hundred watts of electricity (about enough to power an incandescent light bulb) - up to a few thousand watts (enough for a single electric fire) for the largest models.
So - not much power at all, but it generates it for many years on end with no refueling needed and so is ideal for spacecraft that can't be serviced and refueled and are too far from the sun to use solar power or for one reason or another can't use solar power.
But there is hardly any plutonium 238 left for NASA to use as they are no longer refining plutonium. And Russia used to be able to supply it but no longer does.
The US has started creating it again in a small way. Without this then we don't have any technology for the really deep space missions like exploring Pluto, and pretty hard to explore as far as Saturn. NASA's Plutonium Problem Could End Deep-Space Exploration | WIRED
Though the ESA JUICE explorer of Jupiter's moons will use solar panels so you can get that far with especially efficient solar panels.