> In particular, does it have a closed form or notable algorithm for computing it efficiently?

Is an $O(k)$ algorithm efficient enough? If so, here is a C++ implementation:

<!-- language: lang-cpp -->

    unsigned long long sumbincoef( unsigned N, unsigned k ) {
      unsigned long long i = 1, bincoef = 1, sum = 1;
      for( i=1 ; i<=k ; ++i ) {
        bincoef = bincoef * (N-i+1) / i;
        sum += bincoef;
      }
      return sum;
    }

Caution: this can overflow for sufficiently large values of $N$ and $k$.

Since one is summing $N\choose i$ for successive $i$, the relevant recursion relation is simply

$${N\choose i} = {N\choose i-1}\frac{N-i+1}{i}$$

so that each term in the sum

$$\sum_{i=0}^k{N\choose i}$$

is calculated from the preceding term in $O(1)$ time.