It takes the Earth about 365.25 days to orbit the Sun on its current elliptical path. That means it moves about 30 km/s, 2×π×(149,600,000 km)/(1 year).
So, given that, how high could Earth's velocity go before its orbit changes?
About 30km/s; that it to say, the velocity cannot change at all if you wish to maintain the same orbit.
Every circular orbit is associated with exactly one orbital velocity. Every general elliptical orbit is associated with exactly one velocity profile--one specific apoapsis velocity, one specific periapsis velocity, and one specific curve in between.
If you speed the Earth up at all, its orbit will be different.
If the Sun were more massive than it is now, then the Earth would have to move faster to maintain the orbit that it currently has. The formula for the rotational velocity is $v^2$ = (G • M) / R, where v the velocity, G is the gravitational constant, M is the mass of the sun, and R is the radius of the orbit. As an example, if the Sun had four times the mass that it currently has, then the Earth would have to be moving twice as fast as it is now to be in the orbit it currently occupies. Of course if the Sun had four times the mass that it currently has then other properties about it would be very different as well.
You can't pick the velocity and distance from the sun independently: either one fixes the other. A planet is in orbit because the gravitational attraction of the star accelerates it, causing its path to be a curve rather than a straight line. If you make the planet move faster, then it would require more acceleration to keep it in its orbit (consider whirling a ball on a string around your head – make it spin faster and you feel more pull on the string), but the only way to get more acceleration is to be closer.
Then the speed would be much greater when Earth reaches the point closest to the sun since it will have accelerated on its way towards the sun. The orbit would be the "same" but the speed throughout the full orbit would differ dramatically.
As mentioned by Itsme2003, v2 = (G • M) / R, where v is Orbital Velocity, G is the Gravitational Constant, M is the mass of the central body, in this case our Sun, and R is the Radius of Orbit.
Since the mass of the sun isn't changing in this scenario, G and M are both constants, so we can conflate the two into a single value, C:
v2 = C / R
This can be rearranged by multiplying both sides by R to give us:
v2 • R = C
What this shows us is that the Orbital Velocity and the Radius of Orbit are inversely proportional. As one increases, the other decreases, and vice versa.
Therefore, we can say with certainty that if you were to hack into the Earth's properties and change its "Orbital Velocity" value, the radius of the Earth's orbit will also change, taking us closer or farther from the Sun, depending on whether you increase or decrease the value.
This has been explained by others in other answers, but I wanted to show it through the inverse proportion relationship of the mathematical formula.