I found a great deal with an advance Lps ballast and it lists a 14uf cap spanning from neutral to the 240 tap, the highest tap. My 150w advance HPS also requires the same value cap, and said cap is wired from neutral to 277, the highest tap. I was wondering if the ballast (71A0450 Lps) could run sans capacitor or if it would burn up without the cap. Thanks in advance (heh).

Does it indicate if the cap is required when you use the ballast-on some the cap is optional.If the ballast HAS to have the cap-running it without it could cause the ballast to overheat.To be on the safe side-use the cap.It will also reduce the power draw of the ballast.The cap tries to make the ballast look like a high PF load to the powerline-compensates for the reactance of the ballast.

It doesn't specify if you need the cap or not. I prolly will rob the HPS kit of its cap since I prolly won't be using it soon. I was told by another member that it wouldn't matter, so I'm not sure what to do. I wanted to use the Lps ballast for a NEMA head but I was unsure anyway about if the ballast would withstand the heat, yet it appears to have a NEMA bracket built on.

If the capacitor is is connected to the same points as the mains (sothe capacitor is required across 240V terminals and you are using 240V mains), there is no problem omitting it.
But when the mains connection is different (e.g. capacitor required across 240V input, but power connected to the 120V), omitting the capacitor would mean higher load on primary winding and that may overheat the ballast. Some ballasts may be still designed for that, but some are not. And it is the latest, which is in danger.
Otherwise if the 250W HID specify the capacitor across the 277V inputs and this SOX specifies the same capacitance across 240V inputs, you may use the one from the 250W, it will only be bigger than needed.