Using Oil? A quart/1,000 miles

Sometimes you have to run them for a while and they will stop burning. Also, at a shop I worked at we would run'em hard after initial break in to seat the rings. Just like they do on the dyno. If it was babied after break in maybe it needs more time to seat the rings?
Could also be other issues like poor cylinder finish/hone. Valve guide clearance.
The 273 in my 65 Cuda does not burn a drop. I just rebuilt that over the winter and as soon as the cam was broke in I was ripping up the street at 6k.