Label dead ends

CJam, 61 bytes

q_N/{{0f+zW%}4*3ew:z3few::z{e__4=_@1>2%'#e=*"#"='X@?}f%}@,*N*

Try it here.

Explanation

Outline:

    q_N/               Read input lines
        {   }@,*       Perform some operation as many times as there are bytes
                N*     Join lines

Operation:

    {0f+zW%}4*         Box the maze with zeroes
    3ew:z3few::z       Mystical 4D array neighborhood magic.
                       (Think: a 2D array of little 3x3 neighborhood arrays.)

    {                        }f%    For each neighborhood, make a new char:
     e_                                 Flatten the neighborhood
       _4=_                             Get the center tile, C
           @1>2%                        Get the surrounding tiles
                '#e=                    Count surrounding roads, n
                    *                   Repeat char C n times
                     "#"=               Is it "#"? (i.e., C = '# and n = 1)
                         'X@?           Then this becomes an 'X, else keep C.

(Martin saved two bytes, thanks!)

JavaScript (ES6), 110 109 bytes

r=>[...r].map(_=>r=r.replace(g=/#/g,(_,i)=>(r[i+1]+r[i-1]+r[i+l]+r[i-l]).match(g)[1]||"X"),l=~r.search`
`)&&r

1 byte saved thanks to @edc65!

Explanation

Very simple approach to the problem. Searches for each #, and if there are less than 2 #s around it, replaces it with an X. Repeats this process many times until it's guaranteed all the dead-ends have been replaced with Xs.

var solution =

r=>
  [...r].map(_=>                    // repeat r.length times to guarantee completeness
    r=r.replace(g=/#/g,(_,i)=>      // search for each # at index i, update r once done
      (r[i+1]+r[i-1]+r[i+l]+r[i-l]) // create a string of each character adjacent to i
      .match(g)                     // get an array of all # matches in the string
        [1]                         // if element 1 is set, return # (the match is a #)
        ||"X"                       // else if element 1 is undefined, return X
    ),
    l=~r.search`
`                                   // l = line length
  )
  &&r                               // return the updated r

<textarea id="input" rows="10" cols="40">########.....######..#..###
#......#######....#..#..#.#
#.##......#...#####..#..###
#..#####..#....#..#######.#
#......#...#####.....##...#
#..###.#...#...###...#..###
##########.#..#..##..#.##.#
..#......#.######.#..#.#.#.
..#......#.#..#.#.#..#.#.#.
..######.###..##..#########</textarea><br>
<button onclick="result.textContent=solution(input.value)">Go</button>
<pre id="result"></pre>