My neons have been spawning as well! :dnc1: if there are other fish in the tank even the parents they will get eaten, so yes you should probably remove the adults if you have somewhere to keep them.
The water will need to be soft and acidic for the eggs to be viable. Also the eggs and fry are light sensitive so they will need the lights off, i think until they are free swimming (someone please correct me if they need the light off longer).
Oh and if one female hasn't yet dropped her eggs, maybe put her and a male in a breeding net until she drops them and then put them back with the rest of the adults.