Arc Fatigue: Hex's storyline (which was first seeded in the earliest Big Finish stories, several years before he even appeared) is widely considered to have come to a natural end after "A Death in the Family", but the story arc was continued for another eight releases over four years after that.
Archive Panic: Pretty daunting when combined with that of the main series, especially now that it's eclipsed the original in runtime, and that's not even counting all of the spin-offs.
Broken Base: The Dark Eyes anthologies seem to have divided some fans, with some people saying the anthology format lends itself poorly to a series that relies on consistent narrative. Dark Eyes 3 seems to have fixed the complaints however.
Though the first two seasons of the Eighth Doctor's adventures in Big Finish Doctor Who were well-received, the 40th anniversary story "Zagreus", with its long confusing plot and some bizarre moments like Ace being a robot duck, met quite mixed reception meaning many think it's a poorly paced, overly long mess, and a wasted opportunity. The 5th, 6th and 7th Doctors even lampshade this. They treat the events like a performance, and neatly anticipate that the critics will call the whole thing overly long, confusing and derivative. Meanwhile some people feel that Zagreus was a marvellous tribute to 40 years of Doctor Who, down to Big Finish and the Expanded Universe. The Divergent Universe arc also met quite mixed reception, with the long stories, Charley Pollard becoming less likeable, and a lack of character development for C'rizz. The rushed ending to the arc was justified in that due to New Who the 8th Doctor left the Divergent Universe earlier than intended to avoid alienating the fandom, but the 8th Doctor's stories in the regular monthly releases after this weren't as well received. However the New Eighth Doctor Adventures began to win back the fandom.
Although most stories tend to either be praised (A Death in the Family) or loathed (Nekromanteia), there are the occasional stories that have fans equally split praising and deriding that story. The golden example is "The Boy That Time Forgot", and stating that its reaction from fans was "polarizing" would be an understatement. Some fans compared it (worse than) The Twin Dilemma, while others called it one of Paul Magrs' best work (which, don't forget, would put it on par with the excellent Find and Replace). It's up to the individual to interpret and rate the story themselves.
The president of the English Empire Nigel Rochester in "Jubilee" is one of the more disturbing villains in the Big Finish series, being a Dalek-worshipping human in charge of a nightmarishly genocidal and xenophobic empire which attempts to emulate the Daleks in all the worst ways. He remains cheerfully upbeat and affable even when mutilating midgets to fit in his "toy Daleks" or beating his wife for speaking in contractions.
Wendle Marr from "Nekromanteia" happens to be the most despicable and unlikable character from a cast made up entirely of despicable and unlikable characters. He starts off the play by sending a fleet of ships and their crew to their deaths, then has the gall to order the flagship's commander to die as per his company's protocol. When the guy refuses, he has his assistant prepare to destroy the guy's livelihood and the lives of his family. He later receives funds to help improve the horrid quality of life for the workers on his pet project, but decides to instead pocket the money for himself and kill all the workers once it is finished. He is so evil that even the aforementioned ship commander (who tries to rape companion Erimem) is more likable than him, with his assistant killing him and making the ship commander CEO of the company instead after his greed nearly causes the destruction of the whole universe.
Continuity Lock-Out: Several of the later main range releases in the Forge / Hex story arc require you to be up to date with previous releases. The trilogy of "Protect and Survive", "Black and White" and "Gods and Monsters" requires you to have heard the preceding trilogy of "Project: Destiny", "A Death in the Family" and "Lurkers at Sunlight's Edge", and that also needs the previous two "Project" stories ("Twilight" and "Lazarus") plus "The Angel of Scutari", "Arrangements for War" and "Thicker than Water" at the very least (you would ideally have also heard as many of Hex and Evelyn's previous stories as possible). Plus there's the Companion Chronicle "Project: Nirvana", which should ideally be listened to between episodes 1 and 2 of "Black and White".
Foe Yay: The Macqueen!Master tries this with the Eighth Doctor, while at the same time noting that he couldn't stand the Seventh.
Fridge Brilliance: To the above Foe Yay example. Why does the Macqueen!Master like Eight, but can't stand Seven? Because, inadvertently thanks to Eight, the Master got a new regeneration cycle, rather than have to continually possess people. And Seven? In their last on-screen encounter, Seven sort of left the Master to die on the exploding Cheetah Planet. And, benefit of hindsight, the Macqueen!Master probably realized that Eight did try to save him from certain death/endless oblivion in the Eye of Harmony.
The Doctor in "To the Death" wanting to save Lucie from a crashing spaceship.
"A Death in the Family" features Evelyn Smythe sealing away the Word Lord as she died from old age and bad health. On September 26, 2014, Maggie Stables passed away in her sleep after a long battle with illness.
The Dalek Time Controller, who manipulates history for the advancement of the Daleks. They are even smart enough to know when not to kill the Doctor due to the Web of Time. This is well-shown in "The Dalek Generation" (even if not in Big Finish it is attached to that continuity) where the Controller forces the Supreme Dalek to say it will obey him and comes very close into out-manipulating the Doctor through history into helping the Dalek plan to turn 400 planets into copies of Skaro. Unlike Dalek Caan, when hurled through time it doesn't turn against the Daleks in horror at what they have done but merely thinks of how it can enable the Daleks to rule the Universe.
Rescued from the Scrappy Heap: Big Finish have made a habit out of this, resurrecting unliked characters and concepts and showing what strong writing and the unlimited visual effects budget of a sound-only medium can do. Perhaps the greatest example is:
The Sixth Doctor. His tenure on TV is poorly regarded, to say nothing of companions like Mel and Peri. Big Finish took Colin Baker from the least-favourite Doctor of many to the (officially!) most-beloved on audio, and it's almost universally considered that his best episodes are in fact Big Finish audios.
Peri and Mel: Miracles can be worked with strong writing. Add to that a tendency to remember and use the characters' backgrounds (Peri's a botanist, Mel's a programmer, but you wouldn't know it from the series) and characterization beyond Ms. Fanservice and Screaming Woman, and many fans re-evaluated them. It also helped that Nicola Bryant's American accent has improved with age, and they let Bonnie Langford put her Panto experience to good use in the fan-favourite Panto episodes.