This sequence is also shown in a short video walkthrough.
Lara's new treasure room is located off the MAIN HALL, on the left if you're facing the front door.
To get in, first go through the GYM to the SWIMMING POOL.
Find the button behind the diving board.
...to open a secret panel opposite the TREASURE ROOM door. Behind this panel is the switch that opens the TREASURE ROOM door.
Return through the GYM to the MAIN HALL. There's no hurry at this point since the secret panel remains open. However, the TREASURE ROOM door is on a timer. So once you've pulled the switch, you need to hurry to get through that door before it closes.
The easiest way is to throw the switch but don't roll just yet. Wait for the switch to go up again and then immediately pull it once more.
...and start to sprint (Forward and / on the PC or Mac, R2 on the PlayStation). If you've pulled the switch twice, the door will just be starting to open, so you'll have plenty of time to get through.