I've had the problem too, and was probably one of the first (on the forums at least) to detail my story. However, its on 'the other forum'. Basically what dmassott said it true though - its a problem where even new rings and valve seals will NOT likely fix the issue. The problem is with the block itself. I had a full rebuild of all seals, rings, guides, etc... by the Master tech at my local dealership. It burned oil just as bad, if not worse, after that! They then replaced the block (leaving the rebuild head in my car) and since then I burn ZERO oil.
You can try a thicker oil but I would advise against it. I would advise checking your oil EVERY time you fill-up with gas, and topping off as necessary. If you let it get to the state where the oil pressure light comes on, you're starving your engine of oil and as mentioned by HoRRo, you REALLY need to look at replacing the tensioner and possibly the timing chain at this point.
I highly doubt its under warranty if its a 2004, as the time limit (5 years?) is well past.
SOLD - 2004 TSX, PWP on Ebony, 6MT, #00032. 2006 short block put in at 92,000km --> warranties are a good thing!.
Hondata, Tein, Koni, Injen, Unorthodox Racing, Comptech, Powerstop, FLP Maven, Japspeed.
NOW - 2014 Kia Optima SX-T turbo fun family hauler.