Rebol - 1̶6̶4̶ 134

Parses the html using the parse sub-language of Rebol. Checks the first 19 pages (ignoring errors). Thanks to @rgchris for improvements!

s: 0 repeat n 20[parse read join http://stackoverflow.com/questions?sort=featured&page= n[any[thru{>+}copy x to{<}(s: s + load x)]]]s

Test

>> s: 0 repeat n 20[parse read join http://stackoverflow.com/questions?sort=featured&page= n[any[thru{>+}copy x to{<}(s: s + load x)]]]s
== 24150

Explanation

Rebol ignores whitespace, hence you can put it all on one line like that if you choose. PARSE takes two inputs, and the first argument (read join ...) is fairly self-explanatory. But here are some comments on the parse dialect instructions, in a more traditional indentation:

s: 0
repeat n 20 [
    parse read join http://stackoverflow.com/questions?sort=featured&page= n [
        ;-- match the enclosed pattern ANY (zero or more number of times)
        any [
            ;-- seek the match position up THRU (and including) the string >+
            thru {>+}
            ;-- copy contents at the current position up TO (but not including) <
            copy x to {<}
            ;-- (Basically, run some non-dialected Rebol if this match point is reached)
            (s: s + load x)
        ]
    ]
]
;-- evaluator returns last value, we want the value in S
;-- (not the result of PARSE, that's a boolean on whether the end of input was reached)
s