The way technological progress always occurs in such societies: either underground in secret, or in a neighboring society that winds up siphoning off the smart members of the oppressed one.
It is good to keep in mind that this is a self-correcting situation. Any society that outlaws technological progress of a particular type will find itself outmatched by one that doesn't. Bans on technology are temporary at best, because they are an attempt to outlaw new understanding of the world, which is the same thing as trying to outlaw certain styles of thought.
As far as computing technology in particular, a wall will be hit when the advance of any existing technology gets to the point that exponentially finer or more complex measurements and calculations become necessary to make a linear increase in quality.
So, for example, aerospace engineering today requires computers to simulate high-speed and low-speed aerodynamics on a single body. Without this capacity we would never have understood the importance of or the way to design variable shape wings for high-speed and low-speed flight.
That puts a concrete cap on certain military technologies, means you can't have high-speed and long range capabilities in the same craft, etc. and ultimately means that the practical cost to your society of fielding a military force that can compete with a society of otherwise similar capacity will be weighted drastically against you. And that's just one type of airplane.
Discovering optimal designs for submarine screws, anything with a guidance system, and any sort of system that compensates for human physical limitations of accuracy, etc. will be impossible beyond a very primitive level. Such a society will eventually either be wiped out by its neighbors, or overturned in a revolution from within by an underground that is vastly more advanced than the governing faction (most likely in an effort supported by a technologically superior external sponsor).
It is worth mentioning that the inevitable imbalance is an outcome of the outlawing of technology, and this can be the driving force behind any number of awesome stories or games. This can be a very cool plot device.