Which monotone Boolean functions are representable as thresholds on sums?
It was mentioned in the comments that these are the positive threshold functions.

As for other characterizations, I found the following to be interesting. Suppose we have a positive threshold function with decreasing weights $w_1\ge w_2\ge\dots w_n$:
$$f(v_1,\dots,v_n)=1 \iff \sum_i w_iv_i\ge 1.$$
Then in particular the set of inputs $(v_1,\dots,v_n)$ for which $f(\vec v)=1$ is an order ideal of the binary majorization lattice with $2^n$ points, which is studied in


  Donald Knuth, "The Art of Computer Programming", Exercise 109 of Section 7.1.1. 


To put it informally, $f$ is the kind of function where making earlier bits 1 makes $f$ more likely to be 1: so e.g. $f(0,1,1)\le f(1,0,1)\le f(1,1,0)$. That is, "some bits matter more", and to remove redundant isomorphic cases we assume earlier bits matter more.

However, not all such functions are positive threshold functions! That is, just because you've ordered the exam questions from most important to least doesn't mean your pass/fail rule is based on just adding up some scores. 

Indeed, the number of positive threshold functions (with decreasing weights) on $n$ variables is given by the sequence
$$2,3,5,10,27,119,\mathbf{1113},\dots$$
(oeis.org sequence A000617)
whereas the number of such order ideals is
$$2,3,5,10,27,119,\mathbf{1173},\dots$$
(oeis.org sequence A132183)