The Mist-covered Mountains of Home
Yep…hoping to learn this one real soon myself 
Diamantina Drover - not exactly ITM - but I have played it at local sessions to general approval.
Also, Culloden’s Harvest can bring a tear to the most jaded eye.
“McCrimmon’s Lament” - played by Alistair Fraser, or sung by Lizzie Higgins [RIP]
“The Mermaid’s Song”
“Taoimse Im Coladh” - “Planxty”
“Sean O’Dwyer Of The Glen” - sung by John Morran [ ex-“Deaf Shepherd” ] or Mick West.
“Return To Kintail” - Alistair Fraser & Tony McManus
“Mo Mhuirnin Ban” - Mary Bergin
“Sliabh Geal cCua” - Harry Bradley
“Mairin de Barra” - sung by Seamus Begley
“Bruach na Carraige Baine” - also Seamus Begley
Brian Finnegan’s “Last of the Starrs”
I like to play ‘The Blossom & the rain’, by Brian Peters.
http://www.youtube.com/watch?v=uLxfc7l63Pg
I play it slower than on this clip, in B minor on an A whistle
What a beautiful tune. I must check out Brian Peter’s other videos. Ordinarily, I don’t even pay attention to boxes, but clearly I made an error.
I’ll second “She Moves Through The Fair” … “Eibhli Gheal Chiuin ni Chearbhaill” gets a bit of play-time at the moment … “Waggoner’s Lad” is worth a bit of time as well.
I lived with a girl named Suzy Maguire for a while, so “Siobhan ni Dhuibhir” can conjur up some memories … the list goes on 
Personally I’m a fan of “Gabriel’s Oboe” from the movie The Mission.
I really like this 
As pure tunes I’d second Women of Ireland and Roisin Dubh, but as Spancil Hill was mentioned in the original post I have to include Lament of the Three Marys. It hit me the first time I listened to Highland Sessions, and never fails to stir emotion, for me at least.
http://youtu.be/3JEiuM_eHuw
Empire of the Sun’s version of Suo Gan. I missed the beginning of this movie, had no clue what was going on, found this song so captivating, that I watched the movie to the end, and counted the songs so that when they gave the song credits, I could figure out the name of the song. Then I went to my local music store and bought the sheet music. Not being Welsh, I thought it was an Asian song. I love playing this song on my flute.
http://www.youtube.com/watch?v=bs_T8ssB_28&feature=fvst
http://www.youtube.com/watch?v=To8vbhCG2DY&feature=related
(FYI. That kid grows up to be Batman/Bruce Wayne.)
I seem to favour airs that have what Breandan Breathnach calls “inflection” in them, that is, what “legit” musicians call accidentals.
Here’s one, Spailpin a Ruin
http://www.youtube.com/watch?v=onGGxt19ksg
Also My Laggan Love… when I heard Mick O Brien play it on his CD of airs (The Ancient Voice of Ireland) I played it over and over until I had it.
And I really like D minor airs on the pipes such as The May Morning Dew.
Thanks to the link for Amhrán Mhúinse. That’s on my “to do” list.
“Slaibh na mBan” is another one I enjoy. And “The Little Red Lark”.
Mute mentioned “Somewhere Over the Rainbow”. They aren’t all ITM, he reminded me.
Stackridge’s “C’est la Vie”. http://www.youtube.com/watch?v=-R-OQMEPf10
(I hope this is the right link, because I’m barred from access at the moment.)
CEEDEEs tells me that Stackridge are still going - forty years on - and doing tours!
another 2 - both recent compositions :
“Hannah” - by Sarah Bauhan.
“The First Step” - by Julie Fowlis
Fields of Athenry
“The Lament for Rory” by McPeake on their “These Days” CD. I love the low whistle and U.P. on that tune! I tried to find a link to the song, but the closest I could get was this: http://www.mcpeakemusic.com/looklisten.php . The video clip starts out with the beginning of the lament on the low whistle.
I would definitely say Caoineadh Cú Chulainn, the lament from from Riverdance. It’s performed on the pipes but an excellent version on the low whistle can be found: http://www.youtube.com/watch?v=3yj10u9oQiM right there.
Here’s one I enjoy listening to - and maybe more importantly, it’s fun to play on the whistle:
Lament For Heather by John Whelan from the album ‘Stories’:
http://grooveshark.com/#!/s/Lament+For+Heather/2wFA4B?src=5
David of the white rock for me
I like Poules Houpees, aka Crested Hens. You can find a version by Solas on Youtube.