The following will not answer your question but will (likely) give you some idea about the class of IF groups.
Conjecture (Gromov) A hyperbolic group is either virtually free or contains a surface subgroup.
If one believes this conjecture, the class IF is essentially disjoint from the class of hyperbolic groups.
Conjecture. (Flat closing) If $G$ is a CAT(0) group then either $G$ is hyperbolic or it contains $Z^2$.
If one believes this conjecture, CAT(0) groups will be also useless regarding your question.
Conjecture. (Gersten) Suppose that $G$ is a group which admits a finite $K(G,1)$. Then either $G$ contains a Baumslag-Solitar subgroup or it is hyperbolic.
Remark. The Baumslag-Solitar group $BS(p,q)$ either contains $Z^2$ or $|p|=1$ or $|q|=1$. Furthermore, $BS(p,1)$, $p>1$, contains an infinitely generated abelian subgroup. Hence, apart from the case $|p|=|q|=1$, $BS(p,q)$ cannot be IF. Thus, as Yves observed, instead of saying
"do not contain BS subgroups" as Gersten originally formulated, one can equivalently say "do not contain non-virtually cyclic abelian subgroups".
If one weakens the existence of a finite $K(G,1)$ to finite presentability, then, there are several sets of examples
of finitely presented groups which are not hyperbolic but contain no Baumslag-Solitar subgroups. The first such examples were constructed by Noel Brady ("Branched coverings of cubical complexes and subgroups of hyperbolic groups"), other examples were constructed by Brady, Clay and Dani ("Morse theory and conjugacy classes of finite subgroups II") and
by Lodha ("A hyperbolic group with a finitely presented subgroup that is not of type $FP_3$"). The examples by Brady and Brady--Clay--Dani are not IF. I did not have time to check Lodha's examples but I fully expect them not to be IF either.
Question (C.Y.Tang) (Question FP5 in the database
http://www.grouptheory.info/ of group theory problems)
Is there a non-free non-cyclic finitely presented group all of whose proper subgroups are free?
In view of the above, you will not be able to find examples of finitely presentable IF groups which are not surface groups or free groups in the existing literature (or, maybe you can but to prove that they are IF you will have to work hard). In particular, Tarsky monsters mentioned by Yves, do not look all that bad.