Let F be a field and $S\subset \overline{F}$, ($\overline F$ - the algebraic clousre of F)
And $F(S)$ be the field generated by $S$ over $F$.
When $S$ is finite, any element of $F(S)$ can be expressed as a linear combination of elements in $F$ and $S$.
My question: if $S$ is an infinite set, it is still possible to represent any element of $F(S)$ as a finite linear combination of elements in $F$ and $S$?
If $S$ is any set, $F(S)$ consists of polynomial expressions over $F$ with variables in $S$. Linear combinations are not sufficient, even when $S$ is a singleton. These expressions are finite by definition (so they only use a finite set of variables in $S$), infinite expressions are not defined in this context. So when $S = \{s_1,s_2,\dotsc\}$ for example, $s_1 + s_2 + \dotsc$ is not a well-defined object, but for every $n \in \mathbb{N}$ we have $s_1 + \dotsc + s_n$ and also expressions like $s_1 s_2^2 - s_3 s_5 + s_8$ etc. Notice that your assumption $S \subseteq \overline{F}$ is important here, since it guarantess that polynomial expressions form a subfield. In the general case, $F(S)$ is the field of fractions of the algebra of polynomial expressions.
