Jon would probably feel very different about Robb’s will from Stannis’ offer.
“Jon.” Melisandre was so close he could feel the warmth of her breath. “R’hllor is the only true god. A vow sworn to a tree has no more power than one sworn to your shoes. Open your heart and let the light of the Lord come in. Burn these weirwoods, and accept Winterfell as a gift of the Lord of Light.”
When Jon had been very young, too young to understand what it meant to be a bastard, he used to dream that one day Winterfell might be his. Later, when he was older, he had been ashamed of those dreams. Winterfell would go to Robb and then his sons, or to Bran or Rickon should Robb die childless. And after them came Sansa and Arya. Even to dream otherwise seemed disloyal, as if he were betraying them in his heart, wishing for their deaths. I never wanted this, he thought as he stood before the blue-eyed king and the red woman. I loved Robb, loved all of them … I never wanted any harm to come to any of them, but it did. And now there’s only me. All he had to do was say the word, and he would be Jon Stark, and nevermore a Snow. All he had to do was pledge this king his fealty, and Winterfell was his…
You can’t be the Lord of Winterfell, you’re bastard-born, he heard Robb say again. And the stone kings were growling at him with granite tongues. You do not belong here. This is not your place. When Jon closed his eyes he saw the heart tree, with its pale limbs, red leaves, and solemn face. The weirwood was the heart of Winterfell, Lord Eddard always said…but to save the castle Jon would have to tear that heart up by its ancient roots, and feed it to the red woman’s hungry fire god. I have no right, he thought. Winterfell belongs to the old gods.
Accepting Stannis’ offer would have meant taking Winterfell without the leave of the Starks and betraying their gods. If one takes the position that death releases Jon Snow from his vows, then Robb’s will not only gives that consent but makes it a command, and Jon would be under no obligation to burn the weirwood.
However, there’s going to be tension regardless, because there’s going to be Sansa and the army of the Vale, Rickon and the Manderlys, and possibly Bran as well if the most recent episode of the show is any indication.