If I have two functions $f$ and $g$ and $f$ is pointwise smaller or equal to $g$, i.e. $f(x) \leqslant g(x)$, does this imply that $\text{lfp}\ f \leqslant \text{lfp}\ g$ (provided that their least fixed points exist) and if yes, why?
Both functions are monotone and say the domain is a complete lattice.
Recall that the least fixed point of a monotone $f$ is also the least closed point of $f$, i.e., the least $x$ satisfying $f(x)\leq x$. So in your situation, if $x$ and $y$ are the least fixed points of $f$ and of $g$, respectively, then we have $f(y)\leq g(y) =y$. So $y$ is a closed point of $f$, while $x$ is the least closed point of $f$. Therefore $x\leq y$.