Count the Bytes!

C#, 249 270 bytes

u=>{var c=new System.Net.Http.HttpClient().GetStringAsync(u).Result;var i=c.IndexOf("de>",c.IndexOf('"'+u.Split('/')[4]))+3;return System.Text.Encoding.UTF8.GetByteCount(c.Substring(i,c.IndexOf("<",i)-i-1).Replace("&gt;",">").Replace("&lt;","<").Replace("&amp;","&"));};

I've made some assumptions based on observations of the HTML of pages. Hopefully they hold true for all answer posts.

+21 bytes to unencode & to &. My answer failed to count it's own bytes. ;P

Ungolfed:

/*Func<string, int> Lambda =*/ u => 
{
    // Download HTML of the page.
    // Although new System.New.WebClient().DownloadString(u) looks shorter, it doesn't 
    // preserve the UTF8 encoding.
    var c = new System.Net.Http.HttpClient().GetStringAsync(u).Result;

    // Using the answer id from the URL, find the index of the start of the code block.
    // Empirical observation has found that all code blocks are in <code> tags,
    // there are no other HTML tags that end with "de>",
    // and that '>' and '<' are encoded to '&gt;'/'&lt;' so no jerk can put "de>" 
    // before the first code block.
    var i = c.IndexOf("de>", c.IndexOf('"' + u.Split('/')[4])) + 3;

    // Get the substring of the code block text, unencode '>'/'<'/'&' and get the byte count.
    // Again, empirical observation shows the closing </code> tag is always on a new
    // line, so always remove 1 character when getting the substring.
    return System.Text.Encoding.UTF8.GetByteCount(c.Substring(i, c.IndexOf("<", i) - i - 1).Replace("&gt;", ">").Replace("&lt;", "<")..Replace("&amp;", "&"));
};

Results:

Input:  http://codegolf.stackexchange.com/a/91904/30525
Output: 34

Input:  http://codegolf.stackexchange.com/a/91693/30525
Output: 195 (at the time of this answer the first post has 196, but I can't find the 196th 
byte, even counting by hand, so assuming 195 is correct)

Input:  http://codegolf.stackexchange.com/a/80691/30525
Output: 61

Input:  http://codegolf.stackexchange.com/a/91995/58106 (this answer)
Output: 270

jQuery JavaScript, 97 Bytes

console.log((new Blob([$("#answer-"+location.hash.substr(1)+" code").eq(0).text().trim()])).size)

old version 151 Bytes

s=$('#answer-'+location.hash.substr(1)+' code').eq(0).text().trim();c=0;for(i=s.length;i--;)d=s.charCodeAt(i),c=d<128?c+1:d<2048?c+2:c+3;console.log(c)

old version 163 Bytes

s=$('#answer-'+location.hash.substr(1)+' code').first().text().trim();c=0;for(i=0;i<s.length;i++){d=s.charCodeAt(i);c=(d<128)?c+1:(d<2048)?c+2:c+3;}console.log(c);

input ist location and jQuery is active on the site

Java 7, 314 313 bytes

int c(String u)throws Exception{String x="utf8",s=new java.util.Scanner(new java.net.URL(u).openStream(),x).useDelimiter("\\A").next();int j=s.indexOf("de>",s.indexOf('"'+u.split("/")[4]))+3;return s.substring(j,s.indexOf("<",j)-1).replace("&gt;",">").replace("&lt;","<").replace("&amp;","&").getBytes(x).length;}

Shamelessly stolen from @milk's C# answer and ported to Java 7.
NOTE: This assumes all code is in <code> blocks. Currently it won't work with just <pre> tags (but who uses those anyway?.. ;p).

Ungolfed & test cases:

class M{
  static int c(String u) throws Exception{
    String x = "utf8",
           s = new java.util.Scanner(new java.net.URL(u).openStream(), x).useDelimiter("\\A").next();
    int j = s.indexOf("de>", s.indexOf('"'+u.split("/")[4])) + 3;
    return s.substring(j, s.indexOf("<", j) - 1).replace("&gt;", ">").replace("&lt;", "<").replace("&amp;", "&")
        .getBytes(x).length;
  }

  public static void main(String[] a) throws Exception{
    System.out.println(c("https://codegolf.stackexchange.com/a/91904/30525"));
    System.out.println(c("https://codegolf.stackexchange.com/a/91693/30525"));
    System.out.println(c("https://codegolf.stackexchange.com/a/80691/30525"));
    System.out.println(c("https://codegolf.stackexchange.com/a/91995/58106"));
  }
}

Output:

34
195
61
270