I am looking for $(p, N)$ where $p$, $N$ are integers greater than 1 and
$${\sum_{n=1}^{N}n^{p}}=M^{p}$$
where $M$ is an integer.
$p=2$, $N=24$ leading to $M=70$ is the Cannonball problem, and it was shown that if $p=2$ then $N=24$.
Do we know of any other $(p, N)$ that satisfy this, or how can we prove that $p=2$, $N=24$ is the only case?
This seems to be an open problem.
More general equation has no known non-trivial examples, as described in the article On the equation $1^k+2^k+\dots+x^k=y^n$, so we cannot reasonably expect to find examples in your special case. By the article it is at least fully resolved for $k \leq 11$.
In the other direction, special case of yours is also well-known conjecture by Erdős–Moser (see also this slightly newer reference), it is unknown if $1^k+2^k+\dots+(m-1)^k=m^k$ has other solutions than $1^1+2^1=3^1$. So we cannot reasonably expect to find impossibility proof of your case (as it would prove the above conjecture).
Both of the above problems are also discussed in Richard Guy's Unsolved Problems in Number Theory under D7 Sum of consecutive powers made a power.
On a more positive note, there are some pretty smart people around, so maybe someone can find something new. Then however don't forget to claim a progress on (at least one of) the above open problems ;)