Why would Dent's convictions be overturned?

From a legal perspective, it's worth noting that the criminals he's referring to aren't those that Dent has already convicted in the past but the 549 criminals that Dent convinced Commissioner Gordon to arrest in scene 91.

Not only are the criminals involved not yet convicted (far from it) but Dent makes it abundantly clear to the Mayor that the only reason why Judge Surrillo is entertaining this "farce" is because of Dent's personal involvement:

MAYOR : 549 criminals at once?! How did you convince Surrillo to hear this farce?

DENT : She shares my enthusiasm for justice. After all, she is a judge.

MAYOR : Even if you blow enough smoke to get convictions out of Surrillo, you'll set a new record at appeals for quickest kick in the ass.

DENT : It won't matter.

The whole thing is a dramatically risky move (basically a gigantic nuisance suit) and one that seems doomed to failure. Since all parties recognise that the majority of the criminals will be out on the streets within a few months anyway due to lack of evidence, the Mayor states that he is relying on Dent's public popularity to smooth over the obvious recriminations about the wasted time, effort and cost.

No Dent (or if Dent turns out to be dirty), no trials.

MAYOR : The public likes you, Dent. That's the only reason this might fly. But that means it's on you. They're all coming after you, now. Not just the mob... politicians, journalists, cops - anyone whose wallet's about to get lighter. Are you up to it?

(Dent smiles)

You better be. They get anything on you... those criminals will be back on the streets...

Just in addition to the already provided answer whilst Dent committing murder would discredit him amongst his peers and Surillo would likely drop the trial of the 549 prisoners if he were to do something as it would become a very controversial trial if it went ahead had he done something which could result in the disruption of peace and order which the justice system attempts to prevent. If we look ahead to 'The Dark Knight Rises' we have the Dent Act being implemented in light of Harvey Dent being the shining example of justice in Gotham. The Dent Act from what we're told in the film allows criminals to be detained and incarcerated without parole of their conviction which begins with the 549 criminals already detained. When Jim Gordon reveals, or rather Bane, that Harvey is the murderer the public believes Batman to be this Act would essentially mean all those convictions could be called into question as they have been essentially denied the right to parole which although is a matter of State Law could possibly be considered a restriction on the criminals right to a fair trial which would then suggest the Dent Act would be in conflict with the Sixth Amendment which although does not mention parole rights could be interpreted in such a way by the Constitution.

Because of this possible infraction with United States Constitutional Law Gordon, Bruce and Bane will likely be aware of the damage that could be caused by revealing the truth of Harvey Dent's illegal activities and so Bane was able to use that to determine that such a foundation for which the legislation is based off (that of Dent's clean record and credibility) would give them no cause as to have allowed the conviction of the original 549 criminals let alone allow the legislation to go through of which was based off the manner in which the original 549 were tried and convicted. Obviously the city could not immediately release all those prisoners in the manner they get out and would possibly have some restrictions and loopholes they could formulate to keep a proportion of the criminals locked up but this would not work for all of them and those who got out could not be retried for the same conviction even if the court was able to extract evidence on them from another prisoner concerning that conviction unless they received it before to issue a retrial.

The thing with a retrial however if the criminal was testifying against another it would have to be submitted to them beforehand to issue a retrial but as such a confession would likely be a plea bargain to lessen their own sentence by helping convict another there would be no point submitting this before the deception was revealed as the other criminal would already be locked up so there's no point. They would also have to deal with the amount of appeals first in light of unlawful conviction so some of those prisoners could get out either way which is why Gordon keeps up the charade and Bruce appears to become more of a recluse. Bane simply takes it a step further and uses this to his advantage in order to destroy the city's soul and in turn the city itself.