Twilight falling asleep and using Ulquiorra's leg as a headrest. The CMOH aspect comes from the fact that Ulquiorra is a merciless, logical killing machine, and yet he sees no reason to disturb her rest. Instead he stays in the same position unmoving ALL NIGHT LONG so she can be well rested for the next day.
Twilight's tearful apology to Rainbow Dash for orchestrating Mare Do Well. Made all the more awesome by the fact that she had to track Rainbow Dash down in order to issue the apology.
Rainbow Dash accepting the apology, and actually trying to comfort Twilight makes the scene all the more awesome.
The aftermath in the library with Rainbow Dash snuggling up to Twilight in her sleep is even better.
Twilight hugging Ulquiorra and apologizing for being angry at him when he was carrying out Celestia's orders.
Luna hugging Ulquiorra and telling him that he doesn't have to be alone, and that he has friends, is a scene that you can't read without feeling something.
The scene where Twilight goes out of her way to ask Ulquiorra if he'll be her friend is pure feeldom. Made all the more awesome by the fact that he actually accepts her request, and the deal is sealed with a handshake.
This is topped by the exchange between Applejack and Ulquiorra where he more or less states that he regards her as a friend. That fact bears repeating: the Espada of Emptiness tells Applejack that he regards her as his friend.
What about Pinkie? Ulquiorra is by all measures a murderer devoid of any remorse, and even after knowing this fact she still goes out of her way to throw him a party like he's anyone else. How can you not feel the heartwarming from this?
Pinkie attempting to identify with Ulquiorra because of their barren backgrounds would certainly count as a heartwarming moment.
Pinkie in general. In the chapter where she's giving Twilight a back rub it just reads like comedy gold. But you read it you can tell that she's really worried about the course of action Twilight's embarked on, and wants her to remember that she's not alone no matter what.
Luna feeling sorry for Ulquiorra's past in chapter 60, and hugging the stuffing out of him in chapter 60. Goes one step beyond when she compares him to an orphaned child who never got the chance to know love.
Chapter 63 has a fair share of these moments. Whether it's Twilight hugging Applejack in an effort to comfort her, or Big Macintosh getting a fair share of the hugs being given.
Chapter 67 gives us a short scene with Applejack trying to tend to Twilight after accidentally kicking her, before asking if she can spend the night again. Even though she could tell her no, Twilight says she can stay again, so long as she bunks with Spike instead.
In chapter 68 when Diamond Tiara and Silver Spoon suspect something is going on, the ponies they go to first are the CMC.
The entire exchange between Rainbow Dash and Celestia in chapter 69 is one big CMOH from beginning to end. From Rainbow Dash wanting to stay in Celestia's room because she's afraid of the dark, to Celestia deciding to set her work aside to tend to the pegasus when she's having a nightmare, you're guaranteed to go "aww" at least once.
Rainbow Dash and Luna finally making up in chapter 73. Even more heartwarming is that Rainbow Dash initiated the process.
In between all the awesomeness, there's a major CMOH moment in chapter 79, as Spike is comforted by Twilight after a particularly horrible nightmare.
A very subtle moment is demonstrated in chapter 84 as Ulquiorra and Rainbow Dash tend to different duties to maximize effectiveness. Just before this, Dash attempts to initiate a Handshake Substitute with Ulquiorra. Despite having no need to actually do such, Ulquiorra reciprocates the gesture and performs a fist/hoof bump with her.