Resident ROHbot here - I think it'll be a hard thing to predict whether it will be successful at all, never mind overcome TNA. (And even that would take a couple of years, I'd say)
On one hand, although it's not as gloriously awesome as it once was, it still completely blows WWE and TNA out of the water quality-wise... and with the increased production and management changes involved in the deal, it looks like it will only improve. Also, it's fair to say that most wrestling fans are at least aware of what ROH is, (guys like Danielson, Bourne, Punk and McGuinness have and probably always will be associated with the company) so I don't see why many won't give it a chance. I've lost count of people I've seen who've said something along the lines of "I would watch ROH but can't/don't know how", and with TV and free (IIRC) streaming, they have no excuse. And I'm pretty confident that people who have never seen the likes of Chris Hero, Claudio Castagnoli, Davey Richards, Roderick Strong, El Generico, etc do their thing in the ring will be won over.
On the other hand, perhaps not. I mean, although ROH on HDNet was a pretty great show with a decent online following, it did have some REALLY naff episodes in retrospect. Especially around the time they signed Mike Bennett, for a few weeks the show resembled more of a feeder league for the likes of Grizzly Redwood, Andy Ridge, The Bravados, etc. rather than a representation of what Ring of Honor actually is. Furthermore, it will be interesting to see what the casual WWE/TNA fan reaction is to the product.... considering it IS far more heavily centered on "wrestling" and strong-style than sports entertainment, and will likely still look a bit rough around the edges even with a bigger budget.
In the end though, I think it will really test whether TNA's idea of building most of their company around signing/pushing guys who were big names in another decade/company to the limit. Should be interesting. Hopefully the increased competition will force 'em to pull their finger out and try actual revolutionary ideas to better the company, rather than this whole lazy-ass IMPACT WRESTLING smoke and mirrors. I also wonder what will happen with the TNA-ROH talent exchange program now.