Before this devolves into whether or not citizens on the religious right hit more people with their cars than mainstream religious folk, agnostics, or atheists (although, come to think of it, in my area the Amish probably cause a fair number of accidents by driving black horse-and-buggies around after midnight on weekends.

Big partiers, the Amish. I'm not kidding.), back to the main question.
Is the Republican Party being taken over by the Religious Right? In both the sense that the religious right does not now "control" and is unlikely to do so in the near future,
NO. I think their influence has certainly waned since the initial ascendancy of the Christian Coalition.
Having said that, it's pretty clear to anyone who cares to see it that they have a very large amount of influence over the party, in influencing actions from legislation to judicial appointees to executive orders, etc.
As far categorizing what counts as influence, there are certainly non-religious reasons and support for positions taken by the religious right. But issues like stem cell research and complete opposition to abortion (not even in cases of the health of the mother, etc.), the bulk of the reasoning for the "conservative", "GOP" position is religious, and non-religious reasons are often a reflection of religious thought as well, theology with "God" removed from the letter-head.
Both the Dems and the Repubs are influenced by smaller movements of politically like-minded individuals. In the Democrats, these groups are more numerous and more contentious with one another, often based around strident advocacy on a limited number of issues. I laugh when someone says this or that special interest controls the Dems, because we invariably have at least one constituency screaming at us to go the other way. We're pluralism in action.
The Republicans have a smaller number of groups, but they are larger, have more members, and less tension between them. If I had to break it down real basic groups, I'd say Business Lobby, Cultural Conservatives, and Small Government. This is a gross over-simplification, but I think it's something some need to think about: those GOP-opponents who claim the Repubs are all one of those three, and those party members who deny the influence of the groups they don't like. That isn't directed toward anyone here as much as posts where I've seen people claim the GOP is
ALWAYS for smaller gov't in all aspects, etc.
If I had to list those three in order of "influence", it would be as listed above. I don't think influence is as simple as number of voters self-identifying themselves in some category voting this way or that, it involves money, ability to dictate agendas, and all the stuff we usually don't get to see/hear about. I'd put the religious right under the more general cultural conservatives, as a generally more reactionary but principled group, and one certainly worth distinguishing as the largest block within that category. Their hand can be felt wherever there's no
decent explanation for a policy move other than it's popularity among people who have a very specific, limited reading of their religion (in that the priorities of modern voters with a certain political bent are often different from the authors of the religious dogma they claim, i.e. theological schizophrenia by individual Catholics in condemning politicians on either abortion and school choice
or war, the death penalty, and poverty issues; with their political leanings within our system being de facto more decisive than their religious excuse).
I'd argue that the religious right has a large, obvious influence on the GOP, more so than any similar Democratic group (because our groups tend to be individually be fewer in members and more numerous in competitors), but that it is behind Business in influence within the GOP and really a subset of general Cultural Conservatism. It doesn't dictate the GOP agenda, and it won't anytime soon, but pretending like it doesn't "cause" a significant portion of the GOP platform and agenda is misguided.