Based on an XKCD comic, XKCD 1626 which has the image:
Do any of the active duty ICBM's have the ability to actually target the Sun?
Considering that the R-7 booster that is the basis used by Soyuz/Progress was an ICBM, as were the original Atlas and Titan programs boosters, perhaps orbital. But Earth escape, and then towards the Sun?
The delta-v needed from low Earth orbit to a Hohmann transfer orbit with a periapse inside the Sun is actually "just" 21,300 m/s. But there is a better option. A bi-elliptic transfer to just hit a central body is better when the ratio between the orbital radius and the radius of the central body is larger than 4.82. The orbital radius of the Earth divided by the radius of the Sun is larger than 200. Therefore, accelerating up to solar system escape velocity, and then do a minuscule manoeuvre at a great distance is only going to cost 8750 m/s of delta-v from LEO. Even that is impossible for an ICBM. A fly by of Jupiter is even better, and combined with a multi fly hit-the-sun trajectory, it is only going to cost around 6300 m/s.
The theoretically lowest is however not going to cost more than a transfer orbit towards the Moon (3120 m/s). With three Moon flybys, you can escape the Earth-Moon system with a little $v_\text{inf}$. After a long time waiting for the then long synodic period to sync up, you can do a flyby of the Earth-Moon system to increase the $v_\text{inf}$. (You can not increase the $v_\text{inf}$ just with a flyby of a planet, however, you can if it is a two body system. That is the same effect as a binary star flyby, only much weaker.) Repeat this over again several times for the next century, and the warhead would hit the Sun.
3120 m/s is maybe achievable if you replace parts of the warhead with an upper stage, as a regular ICBM has trouble even going into orbit. Even then, major upgrades are required. But at least, it is within the same magnitude.
Of course, why are you going to nuke the Sun? I assume you have a good reason for that. (Kind of not assuming that actually)
Borrowing from Wikipedia's article on modern ICBMs:
One particular weapon developed by the Soviet Union (Fractional Orbital Bombardment System) had a partial orbital trajectory, and unlike most ICBMs its target could not be deduced from its orbital flight path. It was decommissioned in compliance with arms control agreements, which address the maximum range of ICBMs and prohibit orbital or fractional-orbital weapons.
Orbital and fractional-orbital weapons were prohibited by the SALT II agreement in 1979, and the FOBS missile was phased out in early 1983.
Remember that the "B" in ICBM stands for ballistic; other than the boost phase, ICBM flight is largely unpowered and suborbital.
Given that breaking free of Earth's gravity, which would be necessary in order to reach the Sun, requires a higher velocity than attaining Earth orbit, and by implication of the SALT II agreement no current ICBMs are able to attain even Earth orbit, we can conclude that no matter the transfer orbit trickery involved, no current ICBMs have the capability to target the Sun.
No, nowhere near. Hohmannfan's answer describes various time-consuming and complex routes to the sun which take over 3200m/s ∆v beyond that needed to get to LEO, and ICBMs can't generally get to LEO. They are built to go from point A to point B on Earth's in about half an hour, and don't have substantial performance margin beyond that.
This Q&A touches on the topic a little bit.
