When do Kan extensions preserve limits/colimits?

The pointwise left Kan extension of F along Y is a coend of functors $Lan_{Y}(F) = \int^{x}P(Yx,-).Fx$ where each functor $P(Yx,-).Fx$ is the composite of the representable $P(Yx,-):P \to Set$ and the copower functor $(-.Fx):Set \to D$. As a coend (colimit) of the $P(Yx,-).Fx$ the left Kan extension preserves any colimit by each of these functors.

Now the copower functor $(-.Fx)$ is left adjoint to the representable $D(Fx,-)$ and so preserves all colimits, so that $P(Yx,-).Fx$ preserves any colimit preserved by $P(Yx,-)$. Therefore $Lan_{Y}(F)$ preserves any colimit preserved by each representable $P(Yx,-):P \to Set$ for $x \in C$.

If Y is the Yoneda embedding we have $P(Yx,-)=[C^{op},Set](Yx,-)=ev_{x}$ the evaluation functor at x which preserves all colimits, so that left Kan extensions along Yoneda preserve all colimits.

Or if each $P(Yx,-)$ preserves filtered colimits then left Kan extensions along Y preserve filtered colimits.

I think this is all well known but don't know a reference.

$F$ preserving colimits doesn't imply that $\text{Lan}_Y(F)$ preserves colimits, even if all the categories are cocomplete.

Consider, for example, the case $C = D$ and $F = 1_C$. Then the left Kan extension $\text{Lan}_Y(1_C)$ exists if and only if $Y$ has a right adjoint, and if it does exist, it is the right adjoint of $Y$. (This is Theorem X.7.2 of Categories for the Working Mathematician.) Of course, $1_C$ preserves colimits, but right adjoints usually don't.

(From your notation, I guess you're generalizing from the case where $P$ is the category of Presheaves on $C$ and $Y$ is the Yoneda embedding. In that case, as I bet you know, $\text{Lan}_Y(F) = - \otimes F$ not only preserves colimits but has a right adjoint.)