No you don't want to do that. Linseed oil is not a suitable final finish over paint as it doesn't form a hard, durable film. It's only good when it soaks directly into the wood so that it isn't sitting on the surface but is within the wood fibres.
Your best choices to protect paintwork are a varnish or lacquer, both of which are intended to make durable surface films.
The easiest option here would probably be a waterbased polyurethane varnish, I think this should bond well enough to a very lightly sprayed wood surface however I'm not certain so it would be advisable to do a test piece or two to check.
If you would prefer to go with a lacquer (many spraycan clear finishes) you'll need to test for compatibility with the paint you're using as many spray finishes can interact with paints that they were not intended to go over. If there is a clear finish in the same range as the paint you're using that shouldn't be a problem.
Slightly off-topic but I wanted to say a bit about this:
I've seen a linseed oil finish on bare wood, and after 10 or 15 thin coats, allowing each coat to dry, the wood ends up with a glossy finish.
It shouldn't require 10-15 coats to get a glossy finish, that's just masochism ^_^ Using linseed oil you should start to get a decent sheen at about the third or fourth coat (if you don't you're not doing it right) and it's about as good as you can reasonably expect after about 6-8 coats. That number of coats already requires a little more than a week, who wants to be finishing something for longer than two weeks?
If a gloss level higher than this is needed then the piece should really be varnished because as few as three coats can give you a really good shine and this may take as little as a day and a half in suitable conditions. With the added advantage that varnish also produces a surface that's about ten times more resistant to scratches and water!