Colin Baker gets to toy with this in "The Wrong Doctors", where you have two versions of the Sixth Doctor 'onscreen' at the same time. Rumor has it they wanted a third, but the studio would have imploded from the awesome. Bonnie Langford also gets to play with this in "The Wrong Doctors", where two Mels are running around the story at the same time. However, unlike the Sixth Doctors, the two Mels are night and day.
The Eighth Doctor, played by Paul McGann, constantly gets called a "ponce" and Mistaken for Gay. The Eighth Doctor's second outfit, which made it's debut in "Dark Eyes", greatly resembles Paul McGann's outfit in Withnail and I. The leather jacket is a peacoat version of Marwood's jacket.
When Daphne Ashbrook (who played companion Grace in the TV movie) appears, her character immediately decides she wants to shag the Eighth Doctor, with plenty of Refuge in Audacity. A nod to all the controversy over having the Doctor kiss Grace in the first place.
Just like with Ace, Melanie "Mel" Jane Bush's (Sixth and Seventh Doctors) last name was established in other parts of the Doctor Who Expanded Universe.
Lucie calls the Monk "The Meddling Monk", the first time he's ever been officially called that in all of Doctor Who. He has been called that by fans for quite some time.
The titles used for the Seventh Doctor's Lost Stories were not the ones intended by script editor Andrew Cartmel; they had appeared in a largely speculative article in Doctor Who Magazine some years previously, and Big Finish decided to use them as they were what fandom was used to. The exception was the first story, "Thin Ice", which was referred to as "Ice-Time" in the article but used its originally intended title. (For the record, "Crime of the Century"'s Cartmel-preferred title was "Action at a Distance", "Animal" would have been "Blood and Iron" and "Earth Aid" would have been "Bad Destination".)
Awesome, Dear Boy: Big Finish has gained this repuation which is why they are able to nab such big name actors for their productions. Amongst the actors they have gotten for roles include but are not limited too: Sheridan Smith, Jim Carter, Emilia Fox, Bernard Cribbins, Toby Jones, Graeme Garden, Nicola Walker, Ronald Pickup, Simon Pegg, Anthony Head, Derek Jacobi, and David Warner.
Six gets to spend the entire first episode of his first long Story Arc wearing velvet. Colin Baker originally wanted to play Six in black velvet in the TV series, but it was veto'd.
Eight's companions get in a few snide remarks about his poncy hair, notably when Lucie asks if it's even real. Paul McGann wore a wig when he played the Doctor on TV, since he shaved his head after his audition.
The Doctor keeps trying to take his companions to Blackpool, with varying levels of success. The Doctor remarks at one point that the TARDIS just doesn't want to be there. Him trying to take Lucie Miller to Blackpool, and the TARDIS not being able to go there for mysterious reasons, becomes the Story Arc of an entire season.
Fake American: Apart from manyFake American actors in stories taking place in the USA, Frobisher at one point spends almost an entire episode shapeshifted into the Sixth Doctor. Including his voice. This leads to Colin Baker impersonating Canadian Fake American Robert Jezek playing Frobisher pretending to be the Doctor. It is glorious.
Name's the Same: The Alternate Eighth Doctor, a.k.a. Johann Schmidt, is not to be confused with the Marvel Universe's Red Skull, although both of their backstories involved Nazis.
Promoted Fanboy: A little-known actor named David Tennant was thrilled to receive supporting roles by Big Finish in stories set in the Doctor Who universe, before making the next step to playing the Doctor himself. It then went full circle in 2016 when he returned to the role of the Tenth Doctor in the audio series!
Real-Life Relative: From "An Earthly Child" onwards, Jake McGann, who voices Alexander David "Alex" Campbell (Eighth Doctor's great-grandson), is the son of Eighth Doctor actor Paul McGann.
Real Life Writes the Plot: The reason Molly was Demoted to Extra in "Dark Eyes 3" was because Ruth Bradley had filming commitments in LA and it was thought she would have to record what few lines she did have over there. Turns out she got finished ahead of time, but production had reached the point where the scripts couldn't be rewritten to give her a larger role. Ruth was so busy during the production of "Dark Eyes 4", they decided that Eight should meet here forty years after they parted ways and cast a new older actress.
Recursive Adaptation: Big Finish has produced audio versions of the two major Doctor Who stage plays, The Ultimate Adventure (with Colin Baker) and The Seven Keys to Doomsday (with Trevor Martin reprising his stage role as a non-TV Doctor).
Separated-at-Birth Casting: Sorcha Cusack plays an older Molly O'Sullivan in "Dark Eyes 4". She plays it so well that you'd be convinced that she was related to Ruth Bradley.
Sorcha Cusack plays an older Molly in "Dark Eyes 4", due to a case of Real Life Writes the Plot as Ruth Bradley's bungeoining career made it near impossible for her to be available, so the decision was made to make Molly older. And Sorcha gets Mollly's mannerisms down so perfectly you might be forgiven for thinking that Ruth is still playing her.
Trolling Creator: Nicholas Briggs deliberately trolled the fanbase as much as possible in "Dark Eyes", almost diving into the Last Great Time War for just a few minutes before yanking the idea away again.
The Cliffhanger between Zagreus' first appearance and the subsequent episode lasted a year and a half. Close to the end of that release gap, in the audio "Omega", Zagreus is prominently listed in the credits. It's an in-story actor playing Zagreus on a Gallifrey-themed cruise ship. He has one line.
Molly O' Sullivan would've had a larger role in "Dark Eyes 3" if Ruth Bradley didn't have other commitments. Likewise, she was kept out of "Dark Eyes 4" for the same reason, and was eventually recast with another actress.
When Sally Armstrong was conceived by Nicholas Briggs, he intended for her to eventually become the Doctor's companion in a later "Dark Eyes"; however, halfway through the series the writing duties was picked up by Matt Fitton, and Sally perished in "Dark Eyes 3".
This trope is basically what the "Lost Stories" range runs on, realising stories that were intended to be made for the television series but weren't for various reasons.
You Sound Familiar: A number of actors who've appeared in the show have also been in Big Finish (or were in Big Finish before the revival). These include Derek Jacobi, David Tennant, Anthony Head, Bernard Cribbins, Simon Pegg, Toby Jones, Dan Starkey, and Nicholas Briggs.