Pullbacks of stacks coming from Lie groupoids are not always equivalent to Lie groupoids.
Take G=H=R (the real line).  Define F(x)=0 if x≤0 and F(x)=exp(−1/x^2) if x>0.
The pullback is not equivalent to a Lie groupoid in this situation:
the set-theoretical pullback is (−∞,0]⨯(−∞,0]∪{(x,x)|x∈R},
which is clearly not a smooth manifold.

One can guarantee that the pullback is a Lie groupoid by imposing transversality
conditions on the maps involved.
That is, A ⨯_C B is a Lie groupoid if A_0 → C_0 ← B_0 is transversal
and A_1 → C_1 ← B_1 is transversal,
where subscripts 0 and 1 denotes objects and morphisms respectively.
(In fact, this transversality condition guarantees that the pullback
is also a homotopy pullback, which is almost always what one actually wants.)