If new water stains are present, then new water intrusion is happening.

What was done to fix the water damming problem?

Look for new roof/chimney water problems.

Water intrusion has to be prevented, it cannot be camoflaged.

Everything you've done is a bandaid on a gushing artery. It doesn't address the real problem at all. You're seeing "water stains" as the problem, when the real problem is water infiltration. I would highly suspect that the chimney was not properly flashed and that you might have some rotten decking and even support members if this has been going on for a while.

I appreciate the feedback....I should have written that the water infiltration issue has been addressed at the roof/chimney with repairs.

I'm just looking for advice about how to cover the water stains. I've since done some research and perhaps an oil-based or shellac primer should have been used.

I'm resisting the temptation to say that there is a special circle in you-know-where for people who paint brick work.

If you're correct - and there's no reason to believe that you're not - that the water intrusion problems have been eliminated, then the problem was that the bricks were not really dry when the finishes were applied. Bricks can absorb a surprising amount of water. If the surface feels dry to the touch, it does not mean that all the water has evaporated. In addition, running heaters at one level of the house would not have prevented moisture from migrating via capillary action from the bricks above and below.

Probably not what you want to hear, but the coatings now in place will prevent rapid evaporation and the stains will keep appearing until all the water is finally gone. You might try Kilz again, making sure that it's not the water based kind, but I'm not optimistic.

"they painted with a masonry sealer and then BM paint. They started to see some stains come through, so they did use Kilz, as well. However, after 2 coats of kilz and 2 coats of the interior paint, the staining is not any better, in fact it has gotten worse as of today. "

None of these coatings are ging to stop continued moisture migration.
They can (at best) slow some water vapor movement, but cannot halt liquid water movement by capillary action.

If the bricks have a large water load in them, the water will continue to move as both liquid and vapor through the brick towards the heated space.
It can be enough to actually force a film finish off the surface forming bubbles of water vapor or even liquid water behind the finish.
It just a problem tat even after the source of the water is found and blocked can take time to resolve, with few effective options for speeding the resolution.

'Tincture of time' is often the only real 'cure.'

mainegrower, I would have loved to keep the brick, but it was not a pretty brick at all. And didn't seem to fit with the rest of the house. But maybe someone else would think differently. :)

So, brickeye if there's still moisture, should we just keep it as it is and not apply an oil-based or shellac layer? Will the brick dry through all these new layers of paint? Or are we talking about waiting until the paint inevitably pulls away from the brick (ugh)?

Leave it alone for at last a few months.

As little coating as possible,

You want the moisture to escape, not remain trapped by surface finishes.

I cannot tell you how many floors ad plaster walls I have seen ruined after a leak by trying to 'hurry up' and fix the problem.

Plaster with the finish layer blown off, floors that look like washboards (first one way, then the other as they are sanded and then finally dry).

Not everything is NOW.