I'm going to take the liberty of answering your questions in reverse order, if you don't mind.

The warhead might need maintenance, which might include refueling, on occasion. The nuclear charge itself would need refuelling every few million years for U-235 (half life of 703 million years), every few thousand years if it's Pu-239 (half life of 24,110 years), and so on. Fusion is usually done with heavy hydrogen (deuterium, which is stable, or tritium, which would need to be refueled every few years).

The propellant, well, that's a longer story. It really does depend on the missile in question. Both the R-7 Semyorka (first ICBM) and the SM-65 Atlas (first US ICBM) burned highly refined kerosene and liquid oxygen (the same fuel as, say, the Saturn V, or the R-7 derived Soyuz), and the latter boils off, so the tanks have to be refilled every few days. This was a problem: the R-7 took 20 hours to prep for launch, and couldn't be left ready for more than a day. So, in the name of readiness, the USSR and US moved to storable propellants. This is a problem: solid fuels (Space Shuttle boosters, Minuteman) are hard to control precisely, and storable liquid fuels are usually hydrazine, which is highly toxic, and ignites on contact with... Well, a lot of things. The Titan II used Aerozine 50, which is a 50/50 mix of hydrazine and UDMH (a hydrazine derivative) as fuel, and NTO (nitrogen tetroxide, basically liquid oxygen that's liquid at room temperature) as oxidizer. Proton, the Russian heavy lifter which was originally going to be a huge ICBM (like, Tsar Bomba launching ICBM), uses pure UDMH and NTO. Other fuels, like Pentaborane and Oxygen Diflouride, which would have been insanely dangerous to work with, have also been researched. Since these don't boil off, they can sit in tanks for a long time, and when the rocket needs maintenance, it's because the rocket damaged the vehicle, not because the fuel is too low.

TL;DR: As per the most commonly answered answer in the English language, 'it depends'.