The Batman: Animated Series, or more so its first season, was an extension of Tim Burton's movies. Not a continuation, but an extension. Being major fans of the Tim Burton movies, the creators Bruce Timm and Eric Radomski wanted the series to continue in the same style, look, feel and with some of the same characters.
Eric Radomski: The previous incarnations of Batman I'd seen growing up, the Filmation animated version, that series they did with Adam West, they were all just a bunch of goofs. They were dopey versions of a character that could be really strong and dramatic, and when I saw Tim Burton's movie, I thought that was a good way of looking at this character.(...) Literally the first piece that I did [on BTAS] was lights of a city reflected on a wet pavement, and that was also inspired by the drama of Burton's movie. (Animation World Magazine)
Bruce Timm: Thank God for the Tim Burton movie because it was so extremely darker than anybody had seen Batman before in any kind of mass media (Wizard 2006)
But we were actually quite lucky, when that show was being developed we were coming off the heels of the Tim Burton Batman films, which were very dark in tone. (TMT 2010)
Paul Dini: When Tim Burton's Batman came out, that was the way to go (animationarena.com)
Nicholson looks phenomenal and scary and the suit looks good and the darkness and the world I was thinking like wow, this rocks (Anthology)
But we were actually quite lucky, when that show was being developed we were coming off the heels of the Tim Burton Batman films, which were very dark in tone. (TMT 2010)
Paul Dini: When Tim Burton's Batman came out, that was the way to go (animationarena.com)
Nicholson looks phenomenal and scary and the suit looks good and the darkness and the world I was thinking like wow, this rocks (Anthology)
Another thing that transitioned form the Tim Burton's movies was Batman's character - as in the movie, he's someone who stays in the shadows and keeps to himself
Paul Dini: The thought process on Batman was great, here he's a character that doesn't talk much, which is as it should be. I thought the more you can tell stories like that, the better the show's gonna be(Batman Legacy Continues 2004)
Paul Dini: The thought process on Batman was great, here he's a character that doesn't talk much, which is as it should be. I thought the more you can tell stories like that, the better the show's gonna be(Batman Legacy Continues 2004)
Also, Kevin Conroy used the same technique for his Batman voice as Michael Keaton did - he lowered his tone and half whispered
The shape of Batman's eyes is also the same as the shape of the cut out eyeholes in Batman's mask in Batman Returns
Shirley Walker, who worked with Danny Elfman conducting his orchestrations, was hired to provide the score for the series with the same style, and even the main theme from Tim Burton's movies was adapted.
Bruce Timm: Everybody was very impressed with the music that Danny Elfman had done for the first Batman movie and we thought that was definitely the right direction to go, that kind of retro noir mysterioso kind of score(Batman Legacy Continues 2004)
The shape of Batman's eyes is also the same as the shape of the cut out eyeholes in Batman's mask in Batman Returns
Bruce Timm: Everybody was very impressed with the music that Danny Elfman had done for the first Batman movie and we thought that was definitely the right direction to go, that kind of retro noir mysterioso kind of score(Batman Legacy Continues 2004)
DARK DECO
On the audio commentary for the On Leather Wings episode, Bruce Timm and Eric Radomski said that they designed the series by closely emulating the Tim Burton films' "otherworldly timelessness,". They followed the same design of the city presented in Batman Returns, Art Deco, or as they renamed it, Dark Deco and expressionistic shadowing (often painting backgrounds on black cels in order to give the feeling of omnipresent night). As in Burton's movies, there's a feel of the 1930's Industrial America with people wearing hats and driving old cars, mixed with modern and futuristic technology like computers
Alan Burnett (producer) : We called it Dark Deco. Dark Deco world with some computer technology going for it. It's just other dimension of the contemporary world (Batman Legacy Continues 2004)
Paul Dini: In my mind it was sort of like what if the 1939 world's fair had gone on another 60 years or so, and the cool sense of design and stylishness from the Art Deco years was in operation well until the 90's. That way you could have kind of a rounded look to the cars and a stylish look to the buildings, you could have television and computers but everything would have a black and white screen (Batman Legacy Continues 2004)
Alan Burnett (producer) : We called it Dark Deco. Dark Deco world with some computer technology going for it. It's just other dimension of the contemporary world (Batman Legacy Continues 2004)
Paul Dini: In my mind it was sort of like what if the 1939 world's fair had gone on another 60 years or so, and the cool sense of design and stylishness from the Art Deco years was in operation well until the 90's. That way you could have kind of a rounded look to the cars and a stylish look to the buildings, you could have television and computers but everything would have a black and white screen (Batman Legacy Continues 2004)
BATMOBILE
The art deco design of the Batmobile mirrored the movie Batmobile's look with the shields up
It also has the same functions and gadgets
BATCAVE
The look of the Batcave is very similar - it's basically a console and sporadic equipment scattered throughout a gigantic cave
BATWING
The Batwing, as in the movie, is of a shape of the stylized bat, with a small cabin located on top. It was also the movie which originated the term Batwing
GRAPPLING GUN
The Grappling Gun originated in Tim Burton's movies, and was since adopted into comic books and subsequent Batman movies, however the first one to adopt it was The Animated Series
Paul Dini: When we did the animated series, we were inspired in part by the tech employed in the movie (Newsrama.com 2010)
Jerry Ordway: It was from the movie. When I did the movie comic, I was amazed at the various gadgets the prop department created for the film (...) At that time, I also understood that DC was not allowed to use any elements of that specific costume design in the comics, as it was property of the film company. I know they relaxed that rule over the last twenty years, though. (Newsrama.com 2010)
Scott Beatty (The DC Comics Encyclopedia, The Batman Handbook: The Ultimate Training Manual): The Grapnel Gun debuted in the 1989 film, but it didn't really catch on in the comics and become a regular feature in the Bat-Arsenal until it was popularized and used to great effect in Batman: The Animated Series, which debuted just a few years later (Newsrama.com 2010)
JOKER
Throughout his entire history, his outfit consisted of green shirt and orange vest.
The 1989 movie was the first instance where the colors were reversed
The Animated Series followed. It also seems like the frozen laugh lines are present in the early episodes
Also, the 1989 movie originated the idea of Joker's pre-Joker persona being a mobster.
The 1989 movie was also the first one to introduce Joker's real name, Jack Napier, which appeared in two episodes, "Dreams In Darkness" and “Joker’s Wild”
Same sunglasses
PENGUIN
The Penguin is a mirror image of Danny DeVito's version of the character. Also, Batman Returns successfully influenced and changed the color of Penguin's outfit from blue to black in all mediums.
The depiction of Penguin (and Catwoman) however, was studio's decision
CATWOMAN
Just like in Batman Returns, Selina Kyle is blonde. However, it's worth noting that she occasional;y did appear as blonde in the Golden Age
TWO FACE
Just like in the 1989 movie, Harvey Dent/Two Face doesn't appear to be a white Caucasian, but also isn't an African American either. He seems to be of a mixed race, although this is purely a speculation as one must have in mind that his appearance was based on Ralph Bellamy
TIM BURTON CAMEO
In the beginning of the episode called "Off Balance", Batman meets an informant named Twitch on the tower of the statue of Liberty. The informant has black hair, a long face, and has been confirmed by the producers to be "Tim Burton"
MISCELLANEOUS
The punch
The sky
The Cathedral Gargoyles
Gotham Globe
"Souvenir" from Catwoman stuck in the costume
If you noticed any other nods and references to the movies in Batman: The Animated Series, feel free to share them in the Comments