A note to begin: this has very little to do with the differences between Analytic and Continental Philosophy, their possible reconcilation, or anything of this sort. This is only about how theology takes concepts from each.
Quite a lot of modern Protestant theology has recourse to modern continental thought, at least insofar as modern Protestant theology is critically reflective and admits of its philosophical underpinnings. This is an interesting phenomenon in the English-speaking world, as most of the philosophy departments (even at the same universities!) belong to the analytic tradition. The use of analytic philosophy is, in my experience, more often done or conditioned by Roman Catholic theologians (this is admittedly excluding those individuals who are in the and/or category: for example, do Plantiga and Craig do philosophy or theology, and are they philosophers or theologians?).
So as a Protestant whose philosophical background lies primarily in the analytic tradition, I have had several experiences which shed light on the relationship between theology and the two traditions:
First, many theologians hear "analytic theology", think "analytic philosophy", and understand "logical positivism". I grant that at one point this might have been an acceptable inference, however, it is certainly no longer the case. There is a great deal of work to be done in theology so that we as Christians do not simply repeat the animosities of philosophy departments, and so that we can critically appropriate analytic methodology (as well as perhaps making our critical reflection on other philosophical traditions a bit more critical).
Second, still on the topic of analytic theology, if the former connection is not made, it seems that folks hear "analytic theology", think "English-speaking Roman Catholics", and understand "Neo-Thomism". Yeah, so, see above on methodology. Not only that, there are enough historical examples to be had of systematic Protestant thinkers, even those who operated on much the same (philosophical) foundation as Aquinas (for example, the Lutheran Johann Gerhard), that we don't need this sort of blanket category. The Neo-Thomists can remain so, but let's not lump everyone in there, shall we?
Third, in regard to the relationship between theology and philosophy period (I may take some heat for this): there was a time when one was expected to have a solid philosophical foundation (in the sense of a wide understanding of the broad strokes) before even doing theology - this was for a long time the case in European universities, and it is some of the root reason why in America today we have an MDiv which one can begin despite having a Bachelor's degree in something else. With the exception of between 5-10% (and excluding all of the "let's get philosophy out of theology" people, oy), I would not hestitate to claim that most theologians (and pastors) do not adequately understand the intellectual foundations upon which they base their theology. The fact of the matter is, that theology requires the thought-categories and critical reflection of philosophy even to be reasoned discourse - and the attempt to appropriate that which we do not understand is detrimental to the endeavour of theology. This is not something that can be remedied with two semesters of survey classes in a Master's program, either! One can become a good theologian without understanding philosophy, but not only is this very rare, one is also an extremely limited theologian. This is a matter of the intellectual responsibility of the Christian - one may say that this is too much, a prospective pastor can't learn good theology, and philosophy, and the ancient languages, and how to use modern languages, and how to engage with literature (the subject of a post yet to come); I say to that individual that the calling to serve God's people is not preparation to do a professional job, but a calling to this level of preparation for interaction with the Word and the world. Especially in today's society, a lower standard is simply inadequate preparation (to be sure, God uses the inadequate, for we are inadequate to His task even when held to a higher intellectual standard; let us not sin that Grace may abound!)
As Christians, of whom to some extent each is a theologian, let us keep clear the differences between methodology in theology and intellectual tradition in philosophy, let us heed the call to develop ourselves enough to be able to differentiate that which we appropriate, and let us above all learn that our commitment to God is not a snatch-time-around-the-edges or a go-home-at-5 obligation, but rather a 100% release of self to the One who reforms (and re-forms) our very being.
Today With Zwingli
54 minutes ago