I've got two touching moments:

1. Grim Fandango, when you are in the Blue Casket and ask Olivia to read a poem, and she reads "Grim Fandango":

"Grim Fandango

With bony hands I hold my partner,
on soulless feet we cross the floor.
The music stops as if to answer,
an empty knocking at the door.
It seems his skin was sweet as mango,
when last I held him to my breast.
But now we dance this grim fandango,
and will four years before we rest."

http://www.youtube.com/watch?v=vrDqS7zSuo0

2. The original intro to Broken Sword - The Shadow of the Templars. There is completley silence and you hear Stobbarts voice for the first time:

"Paris in the fall, the last months of the year, at the end of the millenium. The city holds many memories for me, of music, of cafes, of love, and of death."

And then the brilliant bass violin (I think) just holds one tone, and then goes down a half tone, and another tone and the rest of the orchestra enters. And then the skyline of Paris. WOW.

http://www.youtube.com/watch?v=206BeQNBzhM

Grim Fandango is definitely up there with the best of them. I had a lump in my throat for sure.

The ending of Syberia 1. My eyes got wet a bit, I admit.
TLJ left me thinking for a while.
The reveal of Cobweb's identity in Azrael's Tear.

Oh, and the conclusion of freeware adventure Prodigal, of course. It masterfully manages to build up hope for your brother's rescue...

The intro the BS1 blew me away. The atmosphere, the voice acting, the setting just sent chills down my spine.

- Black Mirror - There's a journal you find on one of the skeletons in the mines. The person who wrote it got trapped in the mine, and the diary documents his last days. The last few pages detail how he's lost hope of being rescued and simply requests that whoever finds his body will give him a proper burial.
There was something very touching about it; it transformed a cliche - a skeleton in the dark - into a human being with a tragic story.
It's completely unrelated to the plot, yet in my opinion it's the best part of the game.

- Tales of Monkey Island - The battle at the end of episode 5.

Spoiler:

The fact that Guybrush keeps trying to make jokes despite being beaten pretty badly really gets to me. I also love the worried exchanges between him and Elaine - both heart-wrenching and hilarious at the same time.

- Sam and Max Season 3 - Ending credits.

Spoiler:

Another instance that combines funny and sad. The silly background events along with Sam's lonely walk made me smile while getting teary eyed.

- Whispered World - The look on Sadwick's face at the end:

Spoiler:

Sad and maybe disappointed and a little mad.
While I think the ending could've been handled better - more hints leading up to it, maybe even a deeper conversation between the two boys - it was still good enough to move me.