If 
I need to print # one and then # twice, until i get to six times. 
You don't want any array - string[] doors = new string[6];, just loops:
for (int line = 1; line <= 6; ++line) {
  for (int column = 1; column <= line; ++column) {
    Console.Write('#'); 
  }
  Console.WriteLine(); 
}
If you have to work with array (i.e. array will be used somewhere else), get rid of magic numbers:
// Create and fill the array
string[] doors = new string[6];
for (int i = 0; i < doors.Length; i++) 
  doors[i] = "#";
// Printing out the array in the desired view
for (int i = 0; i < doors.Length; i++) {
  for (int j = 0; j < i; j++) {
    Console.Write(doors[j]); 
  } 
  Console.Writeline(); 
}
Please, notice that arrays are zero-based (array with 6 items has 0..5 indexes for them)