This requires some time and math but this is the only way I know of:
Go to a player's profile on basketball-reference.com and look a little below his general info for the "Game Log" tab. Select a season and scroll to the bottom where it will say the player's team record in games played. So if you want to see the team's record when he didn't play, look at the team's overall record that season and subtract.
Example:
Miami went 44-38 overall
Miami went 25-15 when Shaq played, therefore..
Miami went 19-23 without Shaq
Simple
Unfortunately I can't find career record in games played so it's a matter of compiling each season's team record with and without Player A and then adding them all up. Btw it's only available dating back to the '87 season.
Also of interest is the "Splits" tab next to the "Game Log" tab on the player's profile. It shows the player's season stats in wins/losses, at home/away, monthly splits, etc.