I have seen criticism on this subreddit of Pipes' The Russian Revolution for reflecting his own political biases, so I was wondering if the first book of his trilogy is similarly problematic.
Well, I am not sure I can speak for "historians", but I can speak for myself as a historian. (I haven't posted here in, ahem, quite a while. I think my flair is gone, but I was once upon a time flaired for the Soviet Union)
First, let's said aside politics as much as we can for the moment. It sounds like you are basically familiar with that part of the historiography of the Soviet Union (if you aren't, ask and I'll elaborate). I will say that I don't really criticize any book for reflecting political bias per se. All historians have one political bias or another, and it will influence their writing. The question is whether or not the history stands up on its own and whether or not that political bias motivated any particularly interesting or revelatory analyses. In the case of Russia Under the Old Regime I am tempted to say it has not.
Historiographically, Pipes work, this included, represents what is by now considered a somewhat outdated mode of historical analysis. His treatment of the peasants is not particularly well done in my opinion - although in his defense the source base for this topic is relatively thin a lot of the time, at least the kinds of sources that would really benefit such a study. Still, even at the time some of his conclusions on the peasantry were pretty controversial.
Also there is an overarching argument in the book that, to cut a few corners here, the elements of Soviet authoritarianism were born out of Imperial Russian history. He links it to what he calls a "patrimonial mentality" that I think is a little too oversimplified and reductionist for my liking. He also draws attention to the Russian criminal code of 1845 as a direct precursor to Soviet (and other 20th century) authoritarian codes. I am not so convinced that this really stands up to scrutiny.
So, the book was written 40 years ago now. It was therefore written without access to a lot of documents that we now have and without the methodological/theoretical developments in the discipline that have also happened since then. The book is no doubt important, and I think that historians would regard it as such. There are many reasonable elements to it and you could certainly do worse if you are looking to learn something about Imperial Russia. His arguments are intellectually interesting, even if I end up disagreeing with many of them. At the same time, you won't see the thing assigned in a ton of undergraduate courses.
Any Soviet Historian will have read it as a matter of learning their historiography. But that is really the place it most clearly belongs at this point, I think. I would not really recommend the book to someone unless they were interested in it for historiographic reasons. But then, that is true for many books written in the 1970s.