Using a chain for more speeds on a lower sprocket count cassette is no problem.
Using a chain for fewer speeds could affect REAR shifting - since the chain's outer width is greater than recommended for the sprocket spacing. It might not engage sprockets well.

Inner width is the same for 6 to 11 speed chains. Outer width difference between 10 and 11 speed chains is negligibly small compared to the front chainring spacing (and fd cage width). So it should not affect the front shifting, if the chain drops from the front chainrings, it's probably FD setup error (limit screws, cable tension, FD positioning etc). I'd also check the chain length - whether it's optimal.