g | x | w | all
Bytes Lang Time Link
400Python3250928T205431ZAjax1234
nanPHP190319T172519Z640KB
132APL Dyalog Extended190320T195753Zvoidhawk
622Python 3190320T135432ZABridgeT
173JavaScript ES6190318T134303ZArnauld
233Python 2190318T184916ZErik the

Python3, 400 bytes

E=enumerate
def L(x,y,d):
 q=[(x,y,[(x,y)])]
 for x,y,p in q:
  if(x,y)in d and'T'==d[(x,y)]:del d[(x,y)];return p
  q+=[(*t,p+[t])for X,Y in[(1,0),(-1,0),(0,-1),(0,1)]if(V:=d.get(t:=(x+X,y+Y)))and V!='M'and t not in p]
def f(b):
 d,r={(x,y):v for x,r in E(b)for y,v in E(r)},[]
 x,y=[j for j in d if'P'==d[j]][0]
 while any(j not in r for j in d if'T'==d[j])and(V:=L(x,y,d)):r+=V;x,y=V[-1]
 return r

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PHP, 385 335 307 270 256 bytes

function c(&$m,$x,$r=-1,$d=0){if($r<0)$r=array_search(P,$m);$m[$r]=V;if(!in_array(T,$m))die;foreach([$r%$x?$r-1:-1,$m[$r-$x]?$r-$x:-1,$m[$r+$x]?$r+$x:-1,$r%$x<$x-1?$r+1:-1]as$o=>$n)if($n>=0&$m[$n]!=M&$m[$n]!=V){echo LUDR[$o];c($m,$x,$n,$o);}echo RDUL[$d];}

Try it online!

Uses a recursive depth-first search to crawl the map until all treasures are found.

Input is a one-dimensional array map and number of columns, example: c( $map, 13 );.
Output is to STDOUT as L,R,U and D.

Or 242 bytes to output as numbers (but I prefer the letters).

Ungolfed

function c( &$m, $x, $r=-1, $d=0 ) {

    // locate starting room ($r) on first move
    if( $r < 0 )                        
        $r = array_search( 'P', $m );

    // mark square on map ($m) with V to show it has been visited
    $m[ $r ] = 'V';

    // end if no more treasures
    if ( ! in_array( 'T', $m ) )
        exit;

    // generate list of adjacent squares
    $dirs = [
        $r % $x ? $r - 1 : -1,            // Left
        $m[ $r - $x ] ? $r - $x : -1,     // Up
        $m[ $r + $x ] ? $r + $x : -1,     // Down
        $r % $x < $x - 1 ? $r + 1 : -1    // Right
    ];

    // consider valid directions for next move
    foreach ( $dirs as $o => $n )
        // if not a Wall or Mountain and not Visited
        if ( $n >= 0 and $m[ $n ] != 'M' and $m[ $n ] != 'V' ) {
            // display the direction
            echo 'LUDR'[ $o ];
            // and recursively keep crawling
            c( $m, $x, $n, $o );
        }

    // reached a dead end, go back to previous square
    echo 'RDUL'[ $d ];   // display the reverse direction

}

Basic Case

+-+-+-+-+-+-+-+-+-+-+-+-+-+
|P| | | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | |T| | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | |M|M|M|T| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | |T| | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+

DDDDDRUUUUURDDRUURDDRUURDDDLDLLLDRRRRURUUUURDDDDDLRRU

Locked Up

+-+-+-+-+-+-+-+-+-+-+-+-+-+
|P| | | | |M| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | |M| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | |T| | |M| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+

DDRUURDD

No Treasure:

+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|P|
+-+-+-+-+-+-+-+-+-+-+-+-+-+

 (no output)

No way to move

+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M|M|M|M|M|M|M|M|M|M|M|P|
+-+-+-+-+-+-+-+-+-+-+-+-+-+

  (no output)

The world is a giant treasure

+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T|T|T|T|T|T|T|T|T|T|T|T|T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T|T|T|T|T|T|T|T|T|T|T|T|T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T|T|T|T|T|T|P|T|T|T|T|T|T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T|T|T|T|T|T|T|T|T|T|T|T|T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T|T|T|T|T|T|T|T|T|T|T|T|T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T|T|T|T|T|T|T|T|T|T|T|T|T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+

LLLLLLUURDRURDRURDRURDDDLLLLLLLDDRURDRURDRURDRURUUUURDDDDDLRRUUUUURDDDDDRUUUUU

1x1

+-+
|P|
+-+

  (no output)

Back Alley

DDDDDRUUUUURDDDDDRUUUUURDDDDDRUUUUURDDDDDRUUUUURRRRRLLLLLDDRRRRRLLLLLDDRRRRR

Hedge Maze (an original creation)

+-+-+-+-+-+-+-+-+-+-+-+-+-+
|P|M| |T|M| | | |M| |M|M|T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |M| |M| | |M|T|M|T| | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T| | |M| |M| |T|M|M|M| |M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M| | | |M| |M|M| | | |M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|T| | |M| |M| | | | |M| |M|
+-+-+-+-+-+-+-+-+-+-+-+-+-+
|M|M| |T| | |M|M|M|M|M| |T|
+-+-+-+-+-+-+-+-+-+-+-+-+-+

DDRRUURLDDDDLLRRDRRUULRUURURRDDLDDRRRURRUULLUDRRRUDLDDDDR

APL (Dyalog Extended), 132 bytes

Saved 16 bytes by having the function ouput 1 0,0 1,¯1 0,0 ¯1 instead of U,L,D,R. TIO link below decodes output back to letters.

I may write an explanation later, but most of the work is creating a graph in a format that the dfns path function likes, then using it to find successive shortest paths between the start and all treasures. Quite efficient, but because treasures are pathed left-to-right, top-to-bottom instead of being based on proximity, the answers are not optimal. 

{z←×`/d←⍴⍵⋄g←z⍴⊂⍬⋄s t m←(⍸∘∊⍵=⊢)¨'PTM'⋄f←{(⍵∊m)∨0=⍺:⍬⋄g[⍺],←⍵}⋄h←(f`⍨,f)⋄⍬≡t⊣{2h`/⍵⊣2h`⌿⍵}x←d⍴⍳z:⍬⋄⊃,/⊃,/2-/¨{⍸x=⍵}¨¨2(g⌂path,)/s,t}

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Python 3, 766 622 bytes

def f(s,W,H):
 Z=enumerate;I=[("L",-1),("R",1),("D",1j),("U",-1j)];o="";P,M,T=map(lambda t:{x+y*1jfor y,l in Z(s)for x,c in Z(l)if c==t},'PMT');E=D={c:{t:0}for t,c in Z(T)}
 while len(E):
  E={}
  for m,N in[(c+d,D[c])for c in D for _,d in I if not c+d in M]:
   if 0<=m.real<W and 0<=m.imag<H:
    T={t:N[t]+1for t in N if not m in D or not t in D[m]}
    if any(T)or not m in D:
     E[m]=T
  for m in E:
   if m in D:
    D[m].update(E[m])
   else:
    D[m]=E[m]
 for m in P:
  if m in D:
   for g in D[m]:
    while D[m][g]:
     for i,d in I:
      if m+d in D and D[m+d][g]<D[m][g]:
       o+=i;m=m+d;break
 return o

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Fairly certain after this that I will never win code-golf, but contributing because I enjoyed the exercise.

This version works by building a dictionary of reachable locations, with each entry being a dictionary whose keys are a unique ID for the treasure, and whose value is the shortest distance to that treasure. The code supports multiple values for P: for each one it outputs a path that collects the treasures. No optimization is attempted.. it picks the nearest treasure from its current location until all treasures are collected.

JavaScript (ES6),  187 183  173 bytes

Takes input as a matrix of characters. Returns an array of signed integers: -2 = down, -1 = right, 1 = left and 2 = up.

ff=(m,X,Y,p=[])=>/T/.test(m)?(g=u=>m.some((r,y)=>r.some((c,x)=>(r[o=c=='M'|c>u|(h=X-x)*h+(v=Y-y)*v^c!='P'?0:f(m,x,y,1/X?[...p,h+2*v]:p,r[x]=3-~r[x]),x]=c,o)))?O:g(-~u))``:O=p

Try it online! (with the output translated back to L, R, U, D)

How?

This AI:

Python 2, 233 bytes

G=input()
e=enumerate
P=T=M=E=frozenset()
for i,k in e(G):
 for j,l in e(k):exec l+'|={(i,j)}'
G={0}
c=-1
while T-G|G&M|G-E-T:
 G-=G;e=list(P)[0];x=c=c+1;r=()
 while x:m=x%5%4;x/=5;r+=m,;e=~-m%2*~-m+e[0],m%2*(m-2)+e[1];G|={e}
print r

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Very slow. Most of the test cases would probably time out.

MPTLRUD → EMPT1302. Any other characters should be considered to be prefixes, separators or postfixes.

GOLFSCORE/181699