Son on a whim, I bought a second Romulan [Pla] Console - and it seems that yes, each instance of the console applies a separate DoT affect to non-plasma weapons, or boosts (although not to the same degree as tac consoles) plasma native weapon damage.
I have no idea if multiple procs (which do happen) will stack their damage or not, but I'd hope they do. Ditto for the hull/shield heal procs stacking their effects. If anyone knows if the damage stacks, I'd love to find out, because I'm thinking about putting together a vesta Recon Vesta to take advantage of it.
One thing I've heard is that when you have multiple procs on a weapon, they all trigger at the same time, rather than separately- a friend of mine has observed this with the polarized disruptors for instance- and if plasma procs don't stack and display t his behaviour, then obviously it wouldn't be worth running multiple consoles, unless you're running native plasma weapons and using them to boost your damage output.
One thing I do like about the [pla] consoles is that the colour of the plasma from their proc is dependent on the triggering weapon. Antiprotons and the KCB generate red plasma, while polarons make purple plasma, and tetryons make blue- which is actually really nice looking.