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:

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.

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:

unsigned long long sumbincoef( unsigned N, unsigned k ) {
  unsigned long long i, 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.