The other answers approach this from dividing the octave and showing that equal divisions must be irrational. Another way of looking at this is to consider whether we can compose an octave by successive multiplications with a rational number. The result is of course the same: we can't.
Start with the Fundamental Theorem of Arithmetic:
every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
Along with that, we need the definition of irreducible fraction:
Every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers, and b > 0.
Two numbers are "coprime" when they have no prime factor in common. Thus, a rational number can be expressed as the set of prime factors (with exponents) that defines its unique irreducible expression. For example, 81:64 may be expressed as 34 * 2−6. When you multiply ratios, you add the exponents of their prime factors. So the product of 3:2 and 5:4 (2−1 * 31 and 2−2 * 51) is 2−3 * 31 * 51, or 15:8.
You're looking for a ratio R that equally divides an octave into N parts, which means that RN equals 2. Can we identify such ratios?
The classic example is that of the perfect fifth, a ratio of 3:2. Other intervals may be found by raising that ratio to a certain power, adjusting the octave by multiplying or dividing by a power of 2. For example, the major second can be 9:8, which is the square of (3:2)2/2. The major third can be 81:64, which is (3:2)4/4. To generate all the pitches in the circle of fifths, keep multiplying. When you get back to C (which some authors will call B♯), you end up with a pitch slightly higher than seven octaves above than the one you started with. The ratio of those two frequencies is 312:212. You can't arrive precisely at the same pitch class because the prime factorization includes 3 with a non-zero exponent.
By generalizing, we can show that the same is true for every ratio R that is not itself a power of 2. (If R is a power of 2, then you have defined one-tone equal temperament, a system in which there is only one pitch class and in which the base interval is the octave or a multiple thereof, which is not interesting. This is the same as dividing the octave using the first root of 2, which is of course 2.)
Consider ratio R with at least one prime factor P unequal to 2. As with the example of the perfect fifth, every time you multiply a frequency by R, the magnitude of P's exponent in the resulting frequency is greater than it was in the original frequency. The goal is to achieve a result where P's exponent is zero, but each multiplication takes us farther from that result. It is therefore impossible.
Of course, one of the things about equal temperament is that 27/12 is so close to 1.5 that perfect fifths are close enough to pure for most purposes. From the standpoint of ratios, this comes about because 3^12 (531,441) is fairly close in value to 2^19 (524,288). You might find decent approximations by looking for numbers that are similarly close in value to some power of two.
In practice, though, I think people who have explored N-tone equal temperament as an approximation of just intonation have chosen N such that some power of the Nth root of 2 is close in value to 1.25 (the ratio of the just major third). If you're interested in some other interval then you can experiment with values of N to find a close approximation to that interval.
I feel compelled to close with this warning, however: if you have too many divisions of the octave, the system is not useful for human musicians. It's only going to be useful for a computer. If you're looking into such a system as an approximation of variable-pitch just intonation, the programmer (or the program) will have to choose which of the several notes to use. In 53-tone equal temperament, a whole step can be 8/53 or 9/53 of an octave in size. In variable-pitch just intonation, a whole step can be 10:9 ratio or a 9:8 ratio. It's basically the same problem. Why not just program your computer to use variable-pitch just intonation?