I came here since I was looking for a solution too. What I did is remove the battery and the plug, press the start button and while presseing plugged the charger, released the start button and press again and it started!
But when started the fan was on max speed and took very long time to boot!!
my MBP is very old by the way.
If the battery is in, does it startup?
Does the MagSafe connector show a green light when connected?
How about putting the dodgy battery back in? Does it boot then? If so, leave it in until you have the new one, unless it has swelled, in which case leave it out and get the new one ASAP.
The machine ought to boot without a battery, but sometimes they just won't.