Let $A$ and $B$ be two compact metric spaces with $B\subset A$- Moreover, let $T\colon A\to A$ continuous and let $S\colon A\to B$ a continuous surjection with $S\circ T=T\circ S$.
Moreover assume that
- $\exists b\in B$ such that $S^{-1}(\left\{b\right\})$ is not finite
- The topological entropy $h(B,T)<\infty$
- $B=\bigcap_{n\geq 0}S^n(A)$, i.e. $B$ is a global attractor with respect to $S$
We then have, of course, the estimation $h(B,T)\leq h(A,T)$.
Moreover, one estimation is $$ h(A,T)\leq h(B,T)+ \sup_{b\in B}h(S^{-1}(\left\{b\right\}),T). $$
My question is if this maybe implies that $h(A,T)=\infty$.
No, it does not imply that $h(A,T)=\infty$. A simple example is a disk $D$ and a rotation $T$ around the center of the disk. Of course, $T|A$ has zero entropy, but if you consider the radial projection $S$ to a smaller disk $B$ centered at the same point (that is, all in $A\setminus B$ projects to the boundary of $B$, while $S|B$ is the identity), then all your hypotheses are satisfied.
PS: Note that in general the symbol $h(S^{-1}b,T)$ is not defined for the usual entropy, you really need to use one of the (possibly nonequivalent) notions for noncompact sets.