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I don't think so, it's more for marketing purposes and to help smooth the bodywork finish. Carbon Fibre can sometimes not be as smooth as you'd want, and needs sanding down in places, so a paint job can be used to make the surface a consistent finish, rather than switching between smooth resin and bits that have been sanded.
Sometimes the teams have new Chassis' which aren't painted in case they're required, although I can only think of times this has happened in Formula E.
Sponsor logos go on it, because of the sponsors commitments, not the regulations.
A lot of the complex front wings don't get painted, especially with so many small and vital components that are expendable/vulnerable, it makes it not worth painting them, especially as it might infringe the aerodynamic properties of those small parts.
EDIT: Here is the full statement about car liveries.
CAR LIVERY 9.1 The provisions of the Code relating to national colours shall not apply to the Championship. Both cars entered by a competitor must be presented in substantially the same livery at every Event, any significant change to this livery during a Championship season may only be made with the agreement of the Formula One Commission. In order that the cars of each team may be easily distinguished from one another whilst they are on the track, the on board cameras located above the principal roll structure of the first car must remain as it is supplied to the team and the second car must be predominantly fluorescent yellow. In order for drivers to be easily distinguished from one another whilst they are on the track, the crash helmet of each driver must, with the exception of one Event of the driver’s choice, be presented in substantially the same livery at every Event during a Championship season. A change to helmet livery will also be permitted if a driver changes team during a Championship season. 9.2 Each car will carry the race number of its driver as published by the FIA at the beginning of the season or the race number that has been allocated to his replacement under Article 26.1(b)(iii). This number must be clearly visible from the front of the car and on the driver’s crash helmet. Prior to the start of the 2014 World Championship season race numbers will be permanently allocated to drivers by ballot, such numbers must then be used by that driver during every Formula One World Championship Event he takes part in throughout his career in Formula 1. A driver’s career in Formula 1 will be deemed to have ended if he does not participate in an Event for two entire consecutive Championship seasons. Any new drivers, either at the start of or during a season, will also be allocated a permanent number in the same way. The only exception to this allocation process will be for the reigning World Champion who will have the option to use the number one. The number that was previously allocated to him will be reserved for him in subsequent seasons if he does not retain the title of World Champion. 9.3 The name or the emblem of the make of the car must appear on the front of the nose of the car and in either case be at least 25mm in its largest dimension. The name of the driver must appear on the external bodywork and be clearly legible.
Also there's proportionally more bodywork at the rear of the car so I imagine the paint weight is not evenly spread after all. I'm sure car setups are made with this rear weight bias in mind already, but ideally don't you want the perfect racecar to be as close as possible to 50:50 weight distribution? Hence any ability to add extra ballast to the front will help get closer to that ideal. Just from my gaming experience in rFactor though, even with maximum ballast weight placed as front as possible the best you can get is 44:56 weight.
