The case of algebra for a monads is discussed explicitely in Gregory Bird thesis (see theorem 6.9). The case of the categories of algebras for an endofunctor or pointed endofunctor can be deduced from the fact that if $F$ is a (pointed) endofunctor on $C$, then $F$-Alg $\rightarrow C$ obviously satisfies the condition of Beck's monadicity theorem, and the induced monad preserves $\lambda$-filtered colimits if $F$ does. All this works for any regular $\lambda$, even $\omega$.
For the case of coalgebra, Jiří Rosický pointed the key references to me by email:
Theorem: Let $A$ be a $\lambda$-accessible category that admits colimits of $\omega$-chain, and let $F: A \rightarrow A$ be a $\lambda$-accessible endofunctor. Then:
Theorem: If $F$ is a $\lambda$-accessible copointed endofunctor or comonads on a locally $\lambda$-presentable $A$, then:
Indeed, this can be deduced from the corollary above using that (for $\lambda$ an uncountable cardinal) the category of $\lambda$-presentable categories and left adjoint functor between them preserving $\lambda$-presentable objects is closed under $\lambda$-small cat weighted pseudo-limits: The category of $M$-coalgbera for copointed endofunctor $M$ can be constructed as a full subcategory of the category of $M_0$-coalgebra where $M_0$ is the underlying endofunctor of $M$ as the equifier of $Id,v:U \rightrightarrows U$ where $U:M_0\text{-Coalg} \rightarrow C$ is the forgetful functor, and $v$ is the natural transforation which on each $M_0$-coalgebra $X$ is the composite $X \rightarrow M(X) \rightarrow X$.
When $M$ is a comonad this is a bit more complicated as we would like to take the equifier of the two natural transformation $X \rightrightarrows M_0^2(X)= M_0(M_0(X))$ corresponding to the two side of the usual square, but as $F^S$ is not a left adjoint functor we cannot directly conclude using 2-limits of diagram of left adjoint functors.
One has a natural functor $M_0$-coalg to $E$ which sends each $M_0$-algebra to the pair of maps $X \rightrightarrows M_0^2 $ corresponding to square defining $M$-algebras and another functor from $M_0^2$-Coalg to $E$ that sends each $f:X \rightarrow M^2_0(X)$ to $(X,f,f)$. taking the (pseudo)pullback of these two functors give us exaclty the category of $M_0$-coalgebra compatible to the comultiplication of $M$. Both these functor clearly preserves all colimits and $\lambda$-presentable objects, so by the results mentioned above, this category is locally $\lambda$-presentable. Combining this with the case of copointed endofunctor we obtain the results.
as an endofunctor of the category of sets (with the direct image functoriality) is a counter example to the first theorem in the case $\lambda=\omega$. That is the category of $\mathcal{P}_f$ coalgebra is not finitely accessible. For the case of comonads, there seems to be a counter-example in the comments of the questions.