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?