I tried

eqn = D[T[r, \[Theta]], {r, 2}] + 1/r D[T[r, \[Theta]], {r, 1}] + 1/r^2 D[T[r, \[Theta]], {\[Theta], 2}] + \[Lambda]^2 T[ r, \[Theta]] == -\[Lambda] r Sin[\[Theta]]    
eqn1 = FullSimplify[# r^2] & /@ eqn    
eqn2 = eqn1 /. \[Lambda] -> 1.2    
lsg = NDSolve[
  {eqn /. \[Lambda] -> 0, T[0, \[Theta]] == 5, 
   D[T[r, \[Theta]], r] == 0 /. r -> 0},
  T,
  {r, 0, 1}, {\[Theta], 0, 2 Pi},
  Method -> {"MethodOfLines", 
    "DifferentiateBoundaryConditions" -> False}]

But that doesn't do the job. Unfortunately the error messages do not give any hint how to avoid the division by zero. A question for Support?