There is in fact a way to extend the fuel supply for nuclear energy, but it involves making plutonium from non-fissile uranium in breeder reactors. Plutonium is also a key ingredient for nuclear bombs. While this is a known way to produce a longer lived supply of nuclear energy, the dangers of making this fuel are such that the international community frowns strongly upon the use of breeder reactors because of their proliferation issues. (see the Web Links to the left of this answer for more about breeder reactors).
The present type of nuclear plants use uranium and there is a finite amount of this on earth, so as it is used up the amount left reduces and it is not renewed. There are other types of reactor that can breed more fuel (usually plutonium) but these have not been commercially developed.
The nuclear reactor itself is not really a resource, but the Uranium235 fuel that is used is nonrenewable, in that once it is fissioned, it's done, and it can't be used again, and naturally no more fissionable material would be produced in any short period of time. However, it is possible to artificially produce nuclear fuel in a breeder reactor, so nuclear fuel could be classified as a renewable resource, up to the point where all the breeder material itself has been used up. No workable commercial type of reactor has been developed. For the present we have to regard nuclear fuel as non renewable, but if and when uranium runs out, this type of breeder reactor may have to be developed (see below)
The current method of obtaining nuclear energy in nuclear reactors is using enriched uranium (U235 is the fissile isotope). There is a finite amount of this on earth, it cannot be replaced, so it is non-renewable. Other possibilities include breeding fissile plutonium from U238, or U233 from thorium. Both would make more use of the earth's resources, though there are technical difficulties which make these methods unattractive at present. They would still be using non renewable sources, though as there is a lot of depleted uranium available (which is mostly U238), and large deposits of thorium in some countries, this would prolong the use of nuclear reactors beyond the time when the U235 runs out. However we are talking here of 50 years or more in the future, so we have time to solve the technical problems.
Bernard Cohen argues that breeder reactors, using fuel dissolved in the oceans, can supply all of Earth's energy needs for billions of years, even after the sun explodes, which should categorize them as "renewable".
Nuclear fusion:
If fusion can be developed this will make another large source of energy available, and the amount of deuterium available in the oceans would product energy for thousands to billions of years, but still not renewable.
Nuclear fusion, if it ever becomes usable, will have a very large source of deuterium from the oceans. It is not strictly renewable, but probably limitless within the lifespan of humanity.
This comes into the category 'is the sun's energy renewable?'. We regard it as so because we can't imagine the sun failing, but of course it will eventually fail, billions of years from now
