Originally Posted by maxmillan
It will be a lot of work priming and extra coating as the last tenant used really ugly thick colours.
The reason why people will often prime before changing the colour of a wall (from red to green, say) is that primers are white, and they often get that white colour from the white pigment titanium dioxide in the primer. Titanium dioxide is white in colour and has the second highest hide of any pigment used in house paints, second only to black. Consequenly, if you want to paint a red wall green, you may want to prime with a green tinted primer because the titanium dioxide in the primer will hide the red colour better than a coat of green paint (which uses Phthalocyanine Green as the pigment) and doesn't hide as well as the pigment titanium dioxide. Be careful with this as many primers use zinc oxide or even chaulk as their white pigment, and so they look equally white, but simply don't hide worth a crap.
In your case, if you want to paint a burgundy wall white, just buy a top quality white or off-white PAINT (not a primer) like Pratt & Lambert Accolade Velvet or Satin in the F4090 or F4790 tint base, or buy Benjamin Moore Aura and get guaranteed hide in one coat or Sherwin Williams top-of-the-line interior latex paint in a white tint base. Those paints are expensive, but if you opt for cheaper paint, you'll have to put on twice as much to hide the underlying colour, and so you save on labour with the better quality paints.
The way to tell that you're getting complete hide in one coat is by painting the perimeter of your walls with either a sash brush or a 3 inch roller and a piece of sheet metal (to keep the paint off the other side of inside corners). Then fill in with a 7 or 10 inch roller. Allow that paint to dry overnight, and in the morning, look for any "picture framing", which is the increase in colour density around the perimeter of the wall where the two coats overlapped.
If you see any of that picture framing, then you're not getting complete hide in one coat. If you were, then one coat of paint would look exactly the same as two, (or three or four coats) and you would not see any of that picture framing effect on the wall at all. The more of that picture framing you see, the less well your paint hides.
Everything else being equal, the flatter the paint, the better it will hide, but the harder it will be to clean marks off the paint. When I was looking for a paint to use in my apartment block I tried to find the highest gloss paint (for easy cleaning) that would hide the underlying beige colour paint in one coat (for less work) and found that I could do that with Pratt & Lambert Accolade in the "designer white" F4790 tint base. If you're wanting to hide a darker colour in one coat, I'd suggest BM Aura.