[Collecting my sporadic comments to one (hopefully) coherent answer.]
A more general question is as follows: For functors $C\stackrel{F}{\to}D\stackrel{U}{\to}E$ and for an index category $J$ such that $UF$ preserves $J$-limits, when does $F$ preserve $J$ limits?
A useful sufficient condition is that if $U$ creates $J$-limits, then in the above situation $F$ preserves $J$-limits. Proof: Let $T\colon J\to C$ be a functor, and suppose that $\tau\colon \ell\stackrel{\cdot}{\to} T$ is a limiting cone in $C$. Since $UF$ preserves $J$-limits, $UF\tau\colon UF\ell\stackrel{\cdot}{\to} UFT$ is a limiting cone in $E$. As $U$ creates $J$-limits, there is a unique lifting of $UF\tau$ to a cone in $D$, and this cone is a limiting cone. But $F\tau\colon F\ell\stackrel{\cdot}{\to} FT$ is such a lift, and hence we're done.
This condition is quite useful, because many forgetful functors are monadic, and monadic functors create all limits (by their definition on pp. 143--144 of Mac Lane and by Ex. 6.2.2 on p. 142 of Mac Lane, or by Proposition 4.4.1 on p. 178 of Mac Lane--Moerdijk, or really by a comment of Tom Leinster from which I learned this :)).
For example, consider the category of all small algebraic systems of some type. From the AFT, we know that the forgetful functor to $\mathbf{Set}$ has a left adjoint, and it is the content of Theorem 6.8.1, p. 156 of Mac Lane that this forgetful functor is monadic.
Returning to the original question, this means that whenever the category $D$ is one of $\mathbf{Grp}$, $\mathbf{Rng}$, $\mathbf{Ab}$,... and $U\colon D\to \mathbf{Set}$ is the forgetful functor, then for any $J$, $UF$ preserves $J$-limits implies $F$ preserves $J$ limits. In particular, if $UF$ is a representable functor (and hence preserves all limits), then $F$ preserves all limits.
Next, let me try to comment on your motivating examples (the one from Q. 23188 and the one from the 'Edit' part of the current question.)
Regarding your example in Q. 23188: Unfortunately I know nothing of Hopf algebras, so I can't understand all the details of your construction. If I understand correctly, you construct a functor $F\colon\mathbf{Rng}\to\mathbf{Grp}$ whose composition with the forgetful functor $U\colon \mathbf{Grp}\to \mathbf{Set}$ is representable. If this is indeed the case, then by the above $F$ itself preserves all limits.
Finally, regarding your example in the edited question: While I know nothing of dynamical systems, from a quick glance at Terence Tao's blog it seems to me that these are just algebras for the trivial monad $\mathbb{T}=\langle \mathrm{Id}_{\mathbf{Set}},1,1\rangle$ in $\mathbf{Set}$. So, if in your example you construct a functor $F\colon\mathbf{Set}\to\mathbf{Set}^{\mathbb{T}}$ whose composition with the forgetful functor $G^{\mathbb{T}}\colon \mathbf{Set}^{\mathbb{T}}\to\mathbf{Set}$ preserves limits, then your $F$ preserves limits (by Ex. 6.2.2, p. 142 of ML).