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.
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.)