First it reads in the input, and makes it nice and rectangular, with a " " border so we don't have to do any bounds checking. Then it looks back through the rectangle, so it always hits a right-bottom corner. When it hits one of these, it directs itself up, following any +s it meets, and clearing them as it goes (with a space). It stops following when it meets a space. Tested on the five given examples.

First it reads in the input, and makes it nice and rectangular, with a " " border so we don't have to do any bounds checking on the horizontal (cheaper to do the checking on the vertical than to put in the extra line). Then it looks back through the rectangle, so it always hits a right-bottom corner. When it hits one of these, it directs itself up, following any +s it meets, and clearing them as it goes (with a space). It stops following when it meets a space. Tested on the five given examples.

C# - 388 433 bytes

Quick job making good use of LINQ.

using C=System.Console;using System.Linq;class P{static void Main(){var D=("\n"+C.In.ReadToEnd()).Split('\n');int i=D.Length,I,J,k,M=D.Max(m=>m.Length),d,c=0;var E=D.Select(l=>l.PadRight(M+1).ToCharArray()).ToList();for(;i-->0;)for(J=M;J-->0;)if(E[I=i][J]==43)for(d=0,c++;(k=E[I+=(d-2)%2][J+=(d-1)%2])>32;)if(k<44){E[I][J]=' ';d=d%2<1?E[I-1][J]<99?3:1:E[I][J+1]!=45?0:2;}C.WriteLine(c);}} 

using C=System.Console; using System.Linq; class P { // this is far too LINQy static void Main() { // 43 + 00101011 // 45 - 00101101 // 124 | 01111100  // read in var D=("\n"+C.In.ReadToEnd()).Split('\n'); // say no to verticle bounds checking     int i=D.Length,I,J,k, M=D.Max(m=>m.Length), // M is max length of a line d, // d is direction of travel (1 == vertical?, 2 == down/right?) c=0; // c is loop count // max all the lines the same length var E=D.Select(l=>l.PadRight(M+1).ToCharArray()).ToList(); // say no to horizontal bounds checking // consume +s for(;i-->0;) // for each line (bottom to top) for(J=M;J-->0;) // for each char (right to left)   if(E[I=i][J]==43) // right-most lower-most +   for(d=0,c++; // go up, increment counter   (k=E[I+=(d-2)%2][J+=(d-1)%2])>32 // move I and J, then check we havn't hit a space   ;)  if(k<44) // +   {   E[I][J]=' '; // toodles     d=   d%2<1? // currently horizontal, must go vertical   E[I-1][J]<99?3 // can't go up, must go down   :1 // can go up, go up   : // currently verticle, must go horizontal   E[I][J+1]!=45?0 // can't go right, must go left   :2 // can go right, go right   ;   } // output result C.WriteLine(c); } } 

C# - 338 388 433 bytes

Saved a bunch of bytes by changing to a 1-dimensional array.

using C=System.Console;using System.Linq;class P{static void Main(){var D=C.In.ReadToEnd().Split('\n');int z,w=D.Max(m=>m.Length)+1,d,c=0;var E=D.SelectMany(l=>l.PadRight(w)).ToList();for(z=E.Count;z-->1;)if(E[z-1]==43)for(d=1,c++;E[z+=d%2<1?w*d-w:d-2]>32;)if(E[z]<44){E[z]=' ';d=d%2>0?z<w||E[z-w]<99?2:0:E[z+1]!=45?1:3;}C.WriteLine(c);}} 

using C=System.Console; using System.Linq; class P { static void Main() { // read in var D=C.In.ReadToEnd().Split('\n');      int z, // z is position in E w=D.Max(m=>m.Length)+1, // w is width d, // d is direction of travel (1 == horizontal?, 2 == down/right?) c=0; // c is loop count // make all the lines the same length var E=D.SelectMany(l=>l.PadRight(w)).ToList(); // say no to horizontal bounds checking // consume +s for(z=E.Count;z-->1;) if(E[z-1]==43) // right-most lower-most + for(d=1,c++; // go left, increment counter E[z+=d%2<1?w*d-w:d-2]>32 // move z, then check we havn't hit a space (when we do, z is basiclly z - 1)  ;)  if(E[z]<44) // + { E[z]=' '; // toodles d= d%2>0? // currently horizontal, must go vertical z<w||E[z-w]<99?2 // can't go up, must go down :0 // can go up, go up : // currently vertical, must go horizontal E[z+1]!=45?1 // can't go right, must go left :3 // can go right, go right ; } // output result C.WriteLine(c); } } 

C# 433

Quick job hopefully making good use of LINQ

using C=System.Console;using System.Linq;class P{static void Main(){var D=("\n"+C.In.ReadToEnd()+"\n").Split('\n');var E=D.Select(l=>(" "+l.PadRight(D.Max(m=>m.Length+1))).ToCharArray()).ToArray();int L=E.Length,i=L,j,I,J,k,d,c=0;for(;i-->0;)for(j=E[i].Length;j-->0;)if(E[I=i][J=j]==43){d=0;for(c++;(k=E[I+=((d&2)-1)*(d&1)][J+=((d&2)-1)*(~d&1)])>32;)if(k<44){E[I][J]=' ';d=d%2<1?E[I-1][J]<99?3:1:E[I][J-1]!=45?2:0;}}C.WriteLine(c);}} 

using C=System.Console; using System.Linq; class P { // this is far too LINQy static void Main() { // 43 + 00101011 // 45 - 00101101 // 124 | 01111100 // read in var D=("\n"+C.In.ReadToEnd()+"\n").Split('\n'); // say no to verticle bounds checking // max all the lines the same length  var E=D.Select(l=>(" "+l.PadRight(D.Max(m=>m.Length+1))).ToCharArray()).ToArray(); // say no to horizontal bounds checking  // consume +s  int L=E.Length,i=L,j,I,J,k, d, // d is direction of travel (1 == vertical?, 2 == down/right?) c=0; // c is loop count for(;i-->0;)   for(j=E[i].Length;j-->0;)  if(E[I=i][J=j]==43) // right-most lower-most +  {   // use I and J in here   d=0; // go up for(c++; // increment counter (k=E[I+=((d&2)-1)*(d&1)][J+=((d&2)-1)*(~d&1)])>32 // move I and J, then check we havn't hit a space ;) if(k<44) // + { E[I][J]=' '; // toodles d= d%2<1? // currently horizontal, must go vertical E[I-1][J]<99?3 // can't go up, must go down :1 // can go up, go up : // currently verticle, must go horizontal E[I][J-1]!=45?2 // can't go left, must go right :0 // can go left, go left ;   }  } // output result C.WriteLine(c); } } 

C# - 388 433 bytes

Quick job making good use of LINQ.

using C=System.Console;using System.Linq;class P{static void Main(){var D=("\n"+C.In.ReadToEnd()).Split('\n');int i=D.Length,I,J,k,M=D.Max(m=>m.Length),d,c=0;var E=D.Select(l=>l.PadRight(M+1).ToCharArray()).ToList();for(;i-->0;)for(J=M;J-->0;)if(E[I=i][J]==43)for(d=0,c++;(k=E[I+=(d-2)%2][J+=(d-1)%2])>32;)if(k<44){E[I][J]=' ';d=d%2<1?E[I-1][J]<99?3:1:E[I][J+1]!=45?0:2;}C.WriteLine(c);}} 

using C=System.Console; using System.Linq; class P { // this is far too LINQy static void Main() { // 43 + 00101011 // 45 - 00101101 // 124 | 01111100 // read in var D=("\n"+C.In.ReadToEnd()).Split('\n'); // say no to verticle bounds checking    int i=D.Length,I,J,k,  M=D.Max(m=>m.Length), // M is max length of a line d, // d is direction of travel (1 == vertical?, 2 == down/right?) c=0; // c is loop count // max all the lines the same length  var E=D.Select(l=>l.PadRight(M+1).ToCharArray()).ToList(); // say no to horizontal bounds checking  // consume +s  for(;i-->0;) // for each line (bottom to top) for(J=M;J-->0;) // for each char (right to left) if(E[I=i][J]==43) // right-most lower-most + for(d=0,c++; // go up, increment counter (k=E[I+=(d-2)%2][J+=(d-1)%2])>32 // move I and J, then check we havn't hit a space ;) if(k<44) // + { E[I][J]=' '; // toodles d= d%2<1? // currently horizontal, must go vertical E[I-1][J]<99?3 // can't go up, must go down :1 // can go up, go up : // currently verticle, must go horizontal E[I][J+1]!=45?0 // can't go right, must go left :2 // can go right, go right ; } // output result C.WriteLine(c); } }