Are there any combinations of characters in the original Maniac Mansion which make the game impossible to finish?
I don't believe so. Certain combinations will probably make it harder, but it seems you generally only should need 1 character's special skill to beat the game; Dave and the third character can just be bodies to do stuff everyone can do.
I am extrapolating based on info from this walkthrough which has instructions for beating the game using each character's special skill, and simply calls your third character "your third kid."
The walkthroughs do mention to not choose both Syd and Razor (because their skills are the same) but they still don't conclude that the game will be unbeatable with them.
That said, killing off one of your characters could easily make the game impossible.
In Maniac Mansion, you need a single Kid's skill to get past certain obstacles (mainly Purple Tentacle) and you need at least two kids to win for most of the game.
- Dave doesn't have any skills whatsoever
- Jeff can fix the Telephone (to call Edna)
- He doesn't have a skill to get you past Purple Tentacle, so he can't win the game without another Kid's help
- Bernard can fix the Telephone (to call Edna) and Fred's Radio (to call the Meteor Police)
- This gives you the meteor police badge that can get past the Purple Tentacle
- Razor/Syd can record a music cassette which is used to get Green Tentacle's demo tape,
which gets him a recording contract
- Green Tentacle will protect you against Purple Tentacle
- Michael can develop Weird Ed's secret plans, which befriends Weird Ed
- Weird Ed will protect you against Purple Tentacle
- Wendy can fix up the Manuscript written by the Meteor and land him a book contract
- The contract gets you past Purple Tentacle and the Meteor - possibly my favorite ending
So as long as you have one kid with a skill you can get past purple tentacle, and as long as you have one additional kid you can finish the game (for stuff like pushing the gargoyle, prank calling Edna to get to the safe, ringing the doorbell to get into Weird Ed's room etc.)
Dave and Jeff do not have any important skills, and are the only characters who do not have a way of getting past Purple Tentacle. But even if you have Jeff and Dave in your party, your third character will have a way of getting past him. Michael Stum's answer has done a good job of outlining how each of the other characters can get past.
Only Bernard and Jeff are able to fix the phone in the library, which is needed to call Edna. (This is Jeff's only special skill!) But it is possible to distract Edna without the phone, simply by sending a character into her room to get caught, which buys you some time to send in another character while she returns from the dungeon.