No, really, Blackadder does have a heart! From the Blackadder II episode Chains:
Queenie: Did you miss me, Edmund?
Blackadder: Madame, life without you was like a broken pencil.
Queenie: (confused) Explain...?
In "Back and Forth," when Blackadder becomes King of England, he makes Baldrick prime minister, despite the fact that Baldrick is an idiot who annoys the crap out of him. It's almost an Aww, Look! They Really Do Love Each Other moment.
Plus the indication that Blackadder chose to be a good leader, and not a tyrant, as one would half expect from him.
Classic moment from Blackadder Goes Forth
Captain Darling: Don't take that tone with me, Lieutenant, or I'll have you on charge for insubordination!
Lieutenant George: Well I'd rather be on a charge for insubordination than on a charge for deserting a friend!
Captain Edmund Blackadder's Famous Last Words being the first (and perhaps only) ones he's said with any empathy for a human being other than himself.
Blackadder: Good luck, everyone. *Blows whistle to signal going over the top.*
A bit earlier, when Darling arrives with an order to go over the top with everyone else, Blackadder completely ignores the opportunity to taunt his hated pencil pushing nemesis, seeing only a terrified man who'll soon be killed.
He not only doesn't taunt him, he pretends to believe that Darling volunteered to join them. Despite it probably taking him all of two seconds to guess that Darling was pushed into it by his mad superior, he gives Darling the best opportunity to save face and pretend to himself that he's being brave and loyal.
Edmund seems to have leveled up even before the show begins. While he still expresses a combination of contempt and disliking towards George and Baldrick, this seems to be his method of coping with the situation he finds himself in. As a Captain in the British Army he would have had unnumbered opportunities to have them at the very least sent away or locked up on a trumped-up charge, if not have them executed. Even when he finds out that they screwed up and very nearly let him be killed in Corporal Punishment, they're fine in the next episode. It appears that although he doesn't like either George or Baldrick, he secretly enjoys their company to distract from the monotony and terror of the war.
Heck, when George reveals his talent for painting, Blackadder even gives him a genuine, non-cynical compliment.
In Major Star, George is late coming back from his date with General Melchett. Previous Blackadders would have been soundly asleep; this Blackadder stays up waiting for George, worrying aloud about his safety in the trenches.
Brief though it was, the first Blackadder's father finally remembering his son's name.
In episode five of season one, Baldrick reads a letter. A minor thing, but then you have to remember that Baldrick is part of the lower classes, who were not considered eligible for education, and reading was one of them. Either Baldrick taught himself, which seems a bit unlikely as he wouldn't know where to start, or Blackadder and Percy took the time to teach him how to read. Given Blackadder's opinion of the peasants through the episode, that is a noticeable feat.
The fact that in season two Percy was willing to give Edmund all of the money he'd set aside for his old age to save him from the Bishop of Bath and Wells. True, it turned out that Edmund had already stolen and spent the money, but it's the thought that counts.
At the end of that same episode, part of the money Edmund asks for when successfully blackmailing the Bishop is so he can take Baldrick along to Mrs. Miggin's pie shop to celebrate. Uncharacteristically nice for the second incarnation.
In Blackadder Goes Forth, Captain Blackadder, like his predecessors, always puts himself first. However, note how when he can involve George and Baldrick in his escape schemes, he does. Even though they're both idiots who would probably endanger his plans, and George doesn't even want to go. Blackadder would happily leave them if only one person can get away, but the fact that he will save them if he can is pretty sweet, by Blackadder standards.
In Blackadder's Christmas Carol, it's kind of sweet to see Blackadder actually acting like a decent person, especially to Baldrick, even as everyone else fleeces him for everything he's worth. It doesn't last, but still.
Edmund reading a bedtime story for his child wife at the end of "The Queen of Spain's Beard" is surprisingly sweet.
The "some beans" scene in the Second. Blackadder attempts to teach Baldrick to count and is very patient about it, even in the face of Baldrick being incredibly slow to grasp how three and one makes four.