It could be a problem, but it might also be rather useful, in terms of giving the Riverlands a measure of self-government that keeps them happy and loyal rather than resentful and rebellious. After all, the whole point of the Lords Paramount system was to give the Seven Kingdoms an incentive to peacefully assimilate into the new monarchy.
Now, it helps a lot that Houses Tully and Stark are joined by blood – in the short term, it’s going to be a lot easier for Riverlords to maintain their allegiance to a King who’s also Hoster Tully’s grandson, and the Lord Paramount of the Riverlands is less likely to rebel against his nephew than he would with say, King Torrhen Stark. However, as time goes on, it’s probably a good idea for House Stark to marry into House Tully every so often – it keeps them close kin, which makes them more likely to stay loyal, it enhances House Tully’s shaky hold on the Riverlands by giving them an infusion of the royal blood they have never had, and it creates the impression of equality between the two houses.
In the long-term, one of the major factors is going to be how the Lord Paramountcy interacts with the new position of Warden of the Southern Marches. The creation of this post is a strong indication of Robb’s intention to hold onto the Riverlands by right of conquest and acclamation, in part through the projection of royal authority. A Warden of the Southern Marches, after all, would answer directly to the King in the North and have the military authority to act against any threats to his rule, external or internal. In the short-term, choosing Brynden Tully is ideal – Brynden is an excellent military leader who Robb can trust to hold down his southern flank, he’s personally loyal to Robb and will maintain his authority, but he’s also a Tully and thus not an open provocation to either House Tully or Riverlander nationalism.
In the future, however, the Starks would have to be careful about appointments – choosing a Northman could be politically difficult unless that Northman was married into the Riverlands, but choosing a Riverlander could also raise issues of domestic rivalries and that Riverlander’s allegiance to the Starks. Likewise, in terms of how the two offices work, there needs to be a balance between upholding the King’s authority in his southern dominions (giving the office to Edmure’s heirs, for example, would not be a good idea for keeping the North in control of the Riverlands) and not offending the Lord Paramount of the Riverlands. Hence, it might be a good idea have the Wardenship be given to a younger son of House Tully who shows promise in arms and who has been fostered in Winterfell and/or married into the North, to ensure his credentials in both realms.
Ultimately, the longevity of Robb’s new kingdom would depend on how well he manages to balance the interests of both kingdoms. Historically, major political forces in the Riverlands have been willing to accept outside rule as long as they feel that local prerogatives are being respected and that they’re not being used for another region’s interests. It definitely helps that Robb Stark warred to free the Riverlands from Lannister dominion – but in the future, Kings of the North will need to make sure they don’t demand too much, too often to support the North (men for the Wall, military aid in dealing with Wildling invasions), without balancing that with royal support for the Riverlands. So if a Stark King is going to build up a naval port on the Saltspear, as I have recommended, he’d better pair that with building up Seagard; if he’s going to build a canal linking the White Knife to Blazewater Bay, he’d better support a canal on the Blue Fork or linking the Godseye to the Trident.