Tomb Raider
Tombs Guide

In this guide, I have quoted guides for the initial tombs, credit goes to IGN for those. This goes for quoted material only, as well as the video.
Source, and credit for certain parts: [ Register or Signin to view external links. ]

In Tomb Raider, there are a total of 7 optional tombs that you can explore throughout the game. These tombs, always have a puzzle to them, and an award once you complete them. They also go with the story. I would recommend completing these as you are in the area, but if you miss any, or have already completed the story, just use the treasure map and go back to complete them.

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Tombs

Tomb of Unworthy - This is the first tomb you will complete, and cannot be missed as it's part of the story. Located at the Mountain Village

The first task is to get fire onto Lara's torch. Unless you've completed the Base Exterior, you will need to search out sources of fire. There is a firepot in the chamber Lara can easily access. Light things up.

With the lit torch, Lara can burn up the bodies, thus robbing enough weight for the platform to remain airborne a moment longer.

Have Lara push the lift's counterweight off, hop onto the platform, and again towards the climb-able rock wall. The treasure map is at the top of the uneventful climb.

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Hall of Ascension - The second tomb you will explore, if you are going in order. This can be fairly confusing. Located at the Mountain Village.

The biggest puzzle about this tomb is timing the shutters (as they re-open on an invisible timer) and timing Lara's cranking to make sure the platform will "catch" the wind just after Lara clambers onto the raised ground to access the moving platform.

After you close the shutters using their crank, run/jog to the movable platform's crank (the one pictured above) and count out approximately 2 beats (about 2 real-time seconds) and spam crank the platform up; the second windy panel should be open by now.

Scrabble Lara onto the lifted platform and let the wind carry her to the scrabble wall at the end of the track.

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Well of Tears - This tomb is located at the Shanty Town, and can appear to be fairly hidden. "It is located in a house with a strung-up corpse in one of the cult's buildings (just west of the avenue to the gate separating north and south Shanty Town)."

Completing the Tomb is easy, you basically pick up the objects and place them down to lift a ramp, and then use it to set fire to the flammable object.

Stand on the lift, and toss the jerry-cans off the lift. The lift will raise up once Lara removes the cans and leaves the lift, requiring Lara to jump to the lift from the raised ground (where the camp site is) as it moves upward. So, before moving off the platform, plot a course to quickly get Lara back on the raised ground so she can jump onto the moving platform before.
Note that the platform will not stay raised forever; once Lara lands on the platform, you will need to jump from the platform to the ledge to get the treasure chest.

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Chamber of Judgement - The second tomb you may explore in Shanty Town. It can be found below the gondola building.

Lara needs to use the short plank ramp to reach the climb-able rock wall leading to the treasure chest. At least four jerry-cans are needed to keep the ramp upright to give Lara the height she needs to reach the target wall.

Two of the jerry-cans are found on the cave floor, but the third and fourth cans are on fragile wooden platforms that collapse when physically touched. Use a Rope Arrow on the aeroplane tyre to demolish one wood platform for the jerry-can.

And make a simple jump to the wood platform for the fourth jerry-can. Place all four on the ramp (you can drop them by pressing her dodge key or equip a weapon), then attempt your jump.
Once Lara clings to the rock wall with the climbing axe, simply make way to the treasure.

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Stormguard Sanctum - This tomb is the only one located in the Summit Forest area. You will find this later on in the game, after you have some higher bow upgrades. The entrance to the tomb can be found on the north side of the area.

The first obstacle is the green methane gas. Detonate that stuff and quickly climb past where the gas was spewing before the gas fumaroles respawn (because they do).

Past the gas, but before crossing over to mountaineer on the climb-able rock wall, Lara needs to destroy the wooden barricade with a fire arrow (otherwise she cannot get to the treasure). With the obstacle destroyed, you can make way to the treasure without further incident.
Note that the hanging pots are only used for lighting (which you can correct by turning up the gamma or the monitor brightness), and do not really figure into the puzzle much (or at all).

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Flooded Vault - One of two tombs that are located in the Shipwreck Beach area. To enter this tomb, you will need three things. The rope arrow, shotgun, and the pry axe. Although at this point, you should have all of these items.

The starting of the tomb is easy, just follow along until you reach the part that requires a little thinking.

Lara needs to shotgun the wooden barricade for her to get a good spot to pull the raft initially. Be sure to free the raft first by burning its tie/anchor. After pulling the raft close to the other side of the platform, have Lara hop there.

Lara can de-electrify the water just long enough to allow for a brief passage - such as crossing the water between the two flights of flooded stairs. However, any longer trip requires placing the electrical socket onto the insulated raft. To get between the two flights of flooded stairs pull down the counterweight for the socket in the water (i.e. the rope-wrapped chunk of metal hanging above the gap between the two staircases).
Once to the other staircase, Lara can pull the raft into the correct position. When the socket is safely placed on the raft, Lara may reach the treasure without further incident (unless you were stupid enough to push the raft out of position once more).

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Temple of the Handmaidens - This is the second tomb that will be found in the Shipwreck Beach area. To enter the tomb, you will need rope grappler, for opening.

This tomb requires some measure of timing to rope/grapple the second horizontal swing bar. Too early, and the turnstile will snap back before the weighted buoy can interact with it. Shoot too late, and the turnstile will miss the buoy.

Start the puzzle by releasing the weighted buoy. It will float clockwise on a circuitous route, allowing you and Lara to try again if you miss a step later.

The first jump is easy. Wait for the weighted buoy to hit the first turnstile -- this rotates the horizontal swing bar into a position Lara can use to reach the next piece of safe ground.

For the second turnstile, try counting 2 beats (approximately two real-time seconds) and shooting a rope arrow from where Lara lands after the first swing. This usually is the timing for getting the second turnstile into position for the weighted buoy to knock it into place.

For the second horizontal bar, it's best to wait for the bar to swing away, and just beyond the point of Lara can reach it. You need to hit the bar when it goes away from Lara, since the climb-able wall she needs to reach is too far away if the bar is coming back towards her.
With some luck and timing, you will see Lara planting her climbing axe on the wall and can make her way up to the treasure without any further incident.

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