Unlike many of Jane Austen's heroines, Fanny Price isn't feisty, assertive, or especially striking in her personality. She's one of life's outsiders, hanging around the periphery of Mansfield Park and its world, both literally and metaphorically. Fanny has no time whatsoever for the tedious conventions of the Regency marriage market and the arch displays of coquettish flirtatiousness it encourages in young women. This is one game she simply refuses to play.
Whether any of this makes her an unlikable character is entirely a matter of opinion. Certainly, one could well imagine ladies of her milieu being against Fanny on account of her ever so superior airs and ostentatious displays of virtue. Successive generations of literary critics have been even more scathing, railing against what they perceive to be Fanny's insufferable self-righteousness.
But the class-obsessed Jane Austen is at pains to suggest that Fanny's personality is very much a product of her social class. Let's not forget that Fanny is the poor relation to her wealthier, more socially prominent cousins. Fanny herself certainly isn't allowed to forget this, reminded by her rich aunt right from the outset that she is socially inferior:
I should wish to see them very good friends, and would, on no account, authorise in my girls the smallest degree of arrogance towards their relation; but still they cannot be equals. Their rank, fortune, rights, and expectations will always be different. It is a point of great delicacy, and you must assist us in our endeavours to choose exactly the right line of conduct.
Jane Austen's overriding message appears to be this—if Fanny really is unlikable, then it's because her social class is unlikable. Her faults are the faults of her class, in much the same way as her taste for virtue is also a product of her background. Whether we find Fanny likable or not, there's no doubt that her outsider status, with its associated value system, allows her to cast a mordant eye on the often ludicrous conventions and manifest hypocrisies of upper-class life in Regency England.