Fake Nationality: Messed around with. Dr. Chandra is supposed to be Dr. Sivasubramanian Chandrasegarampillai, originally from Madras, but a naturalized US citizen at the University of Illinois Urbana-Champaign. When they cast the Russian-Jewish American Bob Balaban in the role, they dropped any reference to his Indian heritage (and changed his affiliation to the University of Chicago, to boot).
Surprisingly more-or-less averted with the rest of the cast. With the glaring exception of Dana Elcar as Dimitri, most of the Soviets are in fact from the Soviet Union (if not Russia specifically), and any others are Czech. Tanya is a special case: she was played by Helen Mirren, who while English is ethnically Russian (her grandfather was an emigre from the Red October) and can do a flawless Russian accent.
Fake Russian: Dana Elcar as Moiseievich has the most trouble with his accent; his Russian radio messages are dubbed. Helen Mirren's grandfather was Russian (their family was originally named Mironov) and she can pull off the accent, even though she isn't fluent in the language. The rest of the actors playing the Leonov crew are variously really Russian, from other Soviet republics, or Czech.
Freeze-Frame Bonus: In the very first interior shot of the Leonov, the camera tilts down from a closeup of a computer monitor in the medical bay. Printed underneath the monitor is a paragraph of English text, beginning with "You may leave the lavatory if the green exit light is on over the door...." This is actually step 4 of the Zero Gravity Toilet instructions from 2001. Not only do you have to freeze-frame it, you need to watch the movie on Blu-Ray in order to make out the text! Additionally, the image on the monitor is captioned "KIRBUK" in Cyrillic (see Mildly Military on the Main page.)
The space pod that appears in Watto's junkyard in Star Wars Episode I: The Phantom Menace may also have come from 2010's Discovery pod bay, but this has not been confirmed.
Shout-Out: The prop designers for Babylon 5 used the design of the Leonov as inspiration for the Omega Destroyers used by the Earthers on that show. Compare the rotating gravity sections of the two ships, they have a largely identical profile. The designer of the digital prop on B5 assumed that someone would catch his obvious joke and have him fix it, but nobody noticed.
What's odd is that the film version of 2001 gave HAL's creator's name as "Mr Langley", rather than "Dr Chandra". You have to wonder why they didn't simply rename the character Langley when they cast a white guy...