I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might help to begin by skimming the tables of contents of the 18-volume (!) Handbook of Philosophical Logic to get some feeling for what people mean by "philosophical logic." [Edit: The preceding link no longer works; one can find some content using Google Books and the Wayback Machine.] It includes many topics that will likely be unfamiliar to mathematicians, such as temporal logic, multi-modal logic, non-monotonic reasoning, labelled deductive systems, and fallacy theory.
Roughly speaking, philosophical logic is the general study of reasoning and related topics. As in other areas of philosophy, this study is not necessarily formal. However, the success of formal methods in mathematical logic has led philosophers to try to formalize many other kinds of reasoning. Formalized modal logics are perhaps the best known of these. These are not always classified as "mathematical logic" because in mathematics one does not typically reason formally about concepts such as possibility, necessity, belief, etc. On the other hand, once a system of logic has been made sufficiently formal, it can of course be subject to mathematical study. Thus the boundary between (for example) formal modal logic and traditional mathematical logic is somewhat blurry. A notable example of the cross-fertilization that is possible here is Fitting and Smullyan's book on Set Theory and the Continuum Problem, which develops the (highly mathematical) subject of forcing from the perspective of modal logic, providing a fresh and completely rigorous approach to a now-classical mathematical subject.
If I had to summarize in one sentence, I would say that mathematical logic is the subfield of philosophical logic devoted to logical systems that have been sufficiently formalized to admit mathematical study. This is a slightly broader definition of mathematical logic than is customary, but I think it's a good definition in the context of this MO question, which tacitly seems to be asking if mathematicians have anything to learn from so-called "philosophical logic."
Abraham Robinson explicitly referred to Leibniz's idea of infinitesimal quantities when developing non-standard analysis in 1960's. Wikipedia article has a quotation from his book
Robinson, Abraham (1996). Non-standard analysis (Revised edition ed.). Princeton University Press. ISBN 0-691-04490-2.
Added: the idea of expressing logic in an algebraic way is credited to Leibniz; see e.g. the following article in Stanford Encyclopedia of Philosophy:
http://plato.stanford.edu/entries/leibniz-logic-influence/#DisLeiMatLog
Added: Saul Kripke introduced a semantics of possible worlds (really, relational semantics) for modal logic.
http://en.wikipedia.org/wiki/Modal_logic#Semantics
The idea of possible worlds precedes Leibniz, but he devoted a lot of consideration to it. Ironically, his claim that our existing world is the best out of possible ones is perhaps most known from the ridicule it received in Voltaire's "Candide". Oh wait, this is Math Overflow...
Best Answer
Let me mention a few current issues on which I have been involved in the philosophy of mathematics. Of course there are also many other issues on which people are working.
Debate on pluralism. First, there is currently a lively or indeed raging debate on the issue of pluralism in the philosophy of set theory. If one takes set theory as a foundational theory, in the sense that essentially every mathematical argument or construction can be viewed as taking place or modeled within set theory (whether or not it could also be represented in other foundational theories), then the question arises whether set-theoretic questions have determinate answers. On the singularist or universist view, every set-theoretic question has a final, determinate truth value in the one true set-theoretic universe, the Platonic realm of set theory. On the pluralist or multiverse views, we have different conceptions of set giving rise to different set-theoretic truths. Both views are a form of realism, and so the debate breaks apart the question of realism or Platonism from the question of the uniqueness of the intended interpretation.
I discuss these issues at length in my paper
There is a growing literature discussing these issues, part of which you can find here. See also the multiverse tag on my blog.
Potentialism. Another currently active topic of research is the issue of potentialism. This topic arises classically in the idea of potential versus actual infinity, with quite a long history, but current work is looking into various aspects of the classical debate, particularly with respect to considering potentialism as a modal theory.
That is, one separates the potentialist idea from the issue of infinity, and looks upon the potentialism as concerned with the idea of having a family of partial universes, or universe fragments, which can be extended to one another as the universe unfolds.
Øystein Linnebo has emphasized this modal nature to potentialism, and he and I undertook to analyze the precise modal commitments of various kinds of set-theoretic potentialism in our paper:
See also the slides for the tutorial lectures series I gave recently on Set-theoretic potentialism, Winter School in Abstract Analysis 2018.
For an example, I believe that the use of Grothendieck-Zermelo universes in category theory exemplifies the potentialist outlook, since one works inside a given universe until a need arises for a larger universe concept, in which case one freely moves to the larger universe concept.
My work on arithmetic potentialism in
provides a way to understand the philosophy of ultrafinitism, viewing it ultimately as a form a potentialism.